Chapter V: Hand Shadows and how to work them. Illustrated (5)

1 × 9 + 2 = 11
12 × 9 + 3 = 111
123 × 9 + 4 = 1111
1234 × 9 + 5 = 11111
12345 × 9 + 6 = 111111
123456 × 9 + 7 = 1111111
1234567 × 9 + 8 = 11111111
12345678 × 9 + 9 = 111111111

No. XXXVIII.--CHESS CAMEO

BY FRANK HEALEY

_A Masterpiece_

BLACK
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
| |.B.| N |...| |...| |...|
+---+---+---+---+---+---+---+---+
|.P.| |...| |.p.| |.p.| |
+---+---+---+---+---+---+---+---+
| |...| K |...| k |.b.| P |...|
+---+---+---+---+---+---+---+---+
|.Q.| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |.n.| |...| |
+---+---+---+---+---+---+---+---+
WHITE

White to play, and mate in three moves.

DIVINATION BY FIGURES

There is a pleasant touch of mystery in the following method of discovering a person’s age:--Ask any such subjects of your curiosity to write down the tens digit of the year of their birth, to multiply this by 5, to add 2 to the product, to multiply this result by 2, and finally to add the units digit of their birth year. Then, taking the paper from them, subtract the sum from 100. This will give you their age in 1896, from which their present age is easily determined.

No. XXXIX.--CHESS CAMEO

BY FRANK HEALEY

_The “Bristol Prize Problem”_

BLACK
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| n |...| |...| N |.p.| |
+---+---+---+---+---+---+---+---+
| |.N.| |...| |...| Q |...|
+---+---+---+---+---+---+---+---+
|...| b |.k.| P |...| |...| |
+---+---+---+---+---+---+---+---+
| p |...| p |...| |.p.| |...|
+---+---+---+---+---+---+---+---+
|.P.| |.P.| |...| R |...| |
+---+---+---+---+---+---+---+---+
| |...| |.P.| |...| P |.K.|
+---+---+---+---+---+---+---+---+
|.B.| |...| R |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE

White to play, and mate in three moves.

FUR AND FEATHERS

As I came in after a day among the birds and rabbits, the keeper asked me--“Well, sir, what sport?” I replied, “36 heads and 100 feet.” It took him some time to calculate that I had accounted for 22 birds and 14 rabbits.

No. XL.--CHESS CAMEO

BY J. E. CAMPBELL

_Splendid Strategy_

BLACK
+---+---+---+---+---+---+---+---+
| |...| |.N.| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| p |.p.| |
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|.B.| |.k.| p |.P.| |.n.| Q |
+---+---+---+---+---+---+---+---+
| |...| p |.N.| |.P.| |...|
+---+---+---+---+---+---+---+---+
|...| |...| B |...| |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| K |.R.| |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE

White to play, and mate in three moves.

JUGGLING WITH THE DIGITS

The nine digits can be arranged to form fractions equivalent to

1 1 1 1 1 1 1
- - - - - - -
3 4 5 6 7 8 9

thus:--

5823 1 7956 1 2973 1 2943 1 5274 1
----- = - ----- = - ----- = - ----- = - ----- = -
17469 3 31824 4 14865 5 17658 6 36918 7

9321 1 8361 1
----- = - ----- = -
74568 8 75249 9

No. XLI.--CHESS CAMEO

BY W. GRIMSHAW

BLACK
+---+---+---+---+---+---+---+---+
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+---+---+---+---+---+---+---+---+
|.N.| |...| K |...| |...| |
+---+---+---+---+---+---+---+---+
| |.b.| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| k |...| |...| |
+---+---+---+---+---+---+---+---+
| |.P.| p |.p.| p |.P.| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |.P.| P |
+---+---+---+---+---+---+---+---+
| |.Q.| |...| |...| R |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE

White to play, and mate in three moves.

No. XLII.--CHESS CAMEO

BY S. LOYD

BLACK
+---+---+---+---+---+---+---+---+
| |...| |...| |...| P |...|
+---+---+---+---+---+---+---+---+
|...| |.K.| |...| |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| |.p.| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |.p.| |
+---+---+---+---+---+---+---+---+
| |.Q.| |...| |.N.| k |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE

White to play, and mate in three moves.

A MOTOR PROBLEM

This motor problem will be new and amusing to many readers:--

Let _m_ be the driver of a motor-car, working with velocity _v_. If a sufficiently high value is given to _v_, it will ultimately reach _pc_. In most cases _v_ will then = _o_. For low values of _v_, _pc_ may be neglected; but if _v_ be large it will generally be necessary to square _pc_, after which _v_ will again assume a positive value.

By a well-known elementary theorem, _pc_ + _lsd_ = (_pc_)², but the squaring may sometimes be effected by substituting _x_³ (or × × ×) for _lsd_. This is preferable, if _lsd_ is small with regard to _m_. If _lsd_ be made sufficiently large, _pc_ will vanish.

Now if _jp_ be substituted for _pc_ (which may happen if the difference between _m_ and _pc_ be large) the solution of the problem is more difficult. No value of _lsd_ can be found to effect the squaring of _jp_, for, as is well-known, (_jp_)² is an impossible quantity.

No. XLIII.--CHESS CAMEO

BY J. G. CAMPBELL

BLACK
+---+---+---+---+---+---+---+---+
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+---+---+---+---+---+---+---+---+
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+---+---+---+---+---+---+---+---+
| |...| |.k.| N |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |.N.| |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| P |...| B |...|
+---+---+---+---+---+---+---+---+
|...| |...| |.P.| p |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| |.K.| |.B.|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE

White to play, and mate in three moves.

No. XLIV.--CHESS CAMEO

BY FRANK HEALEY

BLACK
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
| |...| |.K.| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |.R.| |...| |...| |
+---+---+---+---+---+---+---+---+
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+---+---+---+---+---+---+---+---+
|...| |.P.| k |...| |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| |.Q.| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE

White to play, and mate in three moves.

A NEAT METHOD OF DIVISION

To divide any sum easily by 99, cut off the two right-hand figures of the dividend and add them to all the others. Set down the result of this in line below, and then repeat this process until no figures remain on the left to be thus dealt with.

Now draw a line down between the tens and hundreds columns, and add all up on the left of it, thus:--

8694 | 32 120 | 78
87 | 26 1 | 98
1 | 13 | 99
| 14 ------------
------------- 121 and 99 over.
8782 and 14 over. In other words, 122.

The last number on the right of the lines shows always the remainder. If this should appear as 99 (as in the second example above), add one to the number on the left.

No. XLV.--CHESS CAMEO

BY BLUMENTHAL AND KUND

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+---+---+---+---+---+---+---+---+
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+---+---+---+---+---+---+---+---+
|...| |...| B |...| |...| |
+---+---+---+---+---+---+---+---+
| p |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|.r.| n |...| |...| |...| |
+---+---+---+---+---+---+---+---+
| k |...| p |...| |...| |.Q.|
+---+---+---+---+---+---+---+---+
|...| |...| N |...| |...| |
+---+---+---+---+---+---+---+---+
| b |.P.| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE

White to play, and mate in three moves.

No. XLVI.--CHESS CAMEO

BY A. F. MACKENZIE

_A Prize Problem_

BLACK
+---+---+---+---+---+---+---+---+
| |...| |...| |.R.| |...|
+---+---+---+---+---+---+---+---+
|...| |.p.| |.B.| |...| Q |
+---+---+---+---+---+---+---+---+
| n |...| |.R.| |...| |.b.|
+---+---+---+---+---+---+---+---+
|.p.| |.k.| P |...| |...| |
+---+---+---+---+---+---+---+---+
| K |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|.n.| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE

White to play, and mate in three moves.

A SMART SCHOOLBOY

The question, “How many times can 19 be subtracted from a million?” was set by an examiner, who no doubt expected that the answer would be obtained by dividing a million by 19. One bright youth, however, filled a neatly-written page with repetitions of

1,000,000 1,000,000 1,000,000
19 19 19
--------- --------- ---------
999,981 999,981 999,981

and added at the foot of the page, “_N.B._--I can do this as often as you like.”

There was a touch of unintended humour in this, for, after all, the boy gave a correct answer to a badly worded question.

No. XLVII.--CHESS CAMEO

BY A. CYRIL PEARSON

BLACK
+---+---+---+---+---+---+---+---+
| |.N.| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| r |.n.| |...| n |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| |.R.| |...|
+---+---+---+---+---+---+---+---+
|...| |.P.| k |...| |...| |
+---+---+---+---+---+---+---+---+
| |...| p |.P.| |...| |...|
+---+---+---+---+---+---+---+---+
|.Q.| |.P.| |.K.| |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| |...| P |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE

White to play, and mate in three moves.

No. XLVIII.--CHESS CAMEO

BY FRANK HEALEY

_Quite a Gem_

BLACK
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |.b.|
+---+---+---+---+---+---+---+---+
|...| |...| n |...| |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|.K.| |...| |...| |.Q.| |
+---+---+---+---+---+---+---+---+
| |.P.| |.k.| b |...| P |...|
+---+---+---+---+---+---+---+---+
|.R.| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
| B |...| |...| |.P.| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE

White to play, and mate in three moves.

LEWIS CARROLL’S SHORT CUT

Here is a very smart and very simple method of dividing any multiple of 9 by 9, from the fertile brain of Lewis Carroll:--Place a cypher over the final figure, subtract the final figure from this, place the result above in the tens place, subtract the original tens figure from this, and so on to the end. Then the top line, excluding the intruded cypher, gives the result desired. Thus:--

36459 ÷ 9 = 4051,0
= 4051.
36459

No. XLIX.--CHESS CAMEO

BY A. BAYERSDORFER

BLACK
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| K |...| |
+---+---+---+---+---+---+---+---+
| |...| |.k.| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |.p.| B |...| |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| Q |...| |...| |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE

White to play, and mate in three moves.

ANOTHER FREAK OF FIGURES

1 × 8 + 1 = 9
12 × 8 + 2 = 98
123 × 8 + 3 = 987
1234 × 8 + 4 = 9876
12345 × 8 + 5 = 98765
123456 × 8 + 6 = 987654
1234567 × 8 + 7 = 9876543
12345678 × 8 + 8 = 98765432
123456789 × 8 + 9 = 987654321

No. L.--CHESS CAMEO

BY J. DOBRUSKY

BLACK
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|.n.| |...| |...| B |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| q |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| k |...| |...| |
+---+---+---+---+---+---+---+---+
| |.R.| N |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| K |.R.| |...| |...| |
+---+---+---+---+---+---+---+---+
| Q |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |.n.| |...| |
+---+---+---+---+---+---+---+---+
WHITE

White to play, and mate in three moves.

DIVINATION BY NUMBERS

Here is one of the methods by which we can readily discover a number that is thought of. The thought-reader gives these directions to his subject: “Add 1 to three times the number you have thought of; multiply the sum by 3; add to this the number thought of; subtract 3, and tell me the remainder.” This is always ten times the number thought of. Thus, if 6 is thought of--6 × 3 + 1 = 19; 19 × 3 = 57; 57 + 6 - 3 = 60, and 60 ÷ 10 = 6.

No. LI.--CHESS CAMEO

BY KONRAD BAYER

BLACK
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
| p |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|.k.| |.K.| |...| |...| |
+---+---+---+---+---+---+---+---+
| b |.p.| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
| |.P.| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |.R.| |
+---+---+---+---+---+---+---+---+
WHITE

White to play, and mate in three moves.

COINCIDENCES

Here is a curious rough rule for remembering distances and sizes:--

The diameter of the earth multiplied by 108 gives approximately the sun’s diameter. The diameter of the sun multiplied by 108 gives the mean distance of the earth from the sun. The diameter of the moon multiplied by 108 gives the mean distance of the moon from the earth.

No. LII.--CHESS CAMEO

BY J. BERGER

BLACK
+---+---+---+---+---+---+---+---+
| |...| |...| |...| B |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| p |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| k |...| |...|
+---+---+---+---+---+---+---+---+
|.B.| Q |...| |...| |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |.p.| |.K.| |
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE

White to play, and mate in three moves.

PERSONAL ARITHMETIC

Says Giles, “My wife and I are two,
Yet faith I know not why, sir.”
Quoth Jack, “You’re ten, if I speak true,
She’s one, and you’re a cypher!”

No. LIII.--CHESS CAMEO

BY H. F. L. MEYER

BLACK
+---+---+---+---+---+---+---+---+
| |...| |...| R |...| |...|
+---+---+---+---+---+---+---+---+
|...| k |...| |...| |...| |
+---+---+---+---+---+---+---+---+
| b |.p.| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| p |...| P |...| |...| |
+---+---+---+---+---+---+---+---+
| |.K.| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
| |.B.| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| Q |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE

White to play, and mate in three moves.

DIVISION BY SUBTRACTION

Here is a curious and quite uncommon method of dividing any multiple of 11 by 11.

Set down the multiple of 11, place a cypher under its last figure, draw a line, and subtract, placing the first remainder under the tens place. Subtract this from the next number in order, and so on throughout, adding in always any number that is carried. Thus:--

363 56408 375034
0 0 0
--- ----- ------
33 5128 34094

No. LIV.--CHESS CAMEO

BY FRANK HEALEY

BLACK
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|.B.| |...| |...| |...| p |
+---+---+---+---+---+---+---+---+
| |.r.| p |.p.| |...| b |...|
+---+---+---+---+---+---+---+---+
|...| |.k.| n |...| |.Q.| |
+---+---+---+---+---+---+---+---+
| P |...| |...| p |...| |...|
+---+---+---+---+---+---+---+---+
|.K.| |...| |.P.| |...| N |
+---+---+---+---+---+---+---+---+
| B |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE

White to play, and mate in three moves.

LUCK IN ODD NUMBERS

Perhaps the old saying, “there is luck in odd numbers,” may have some connection with the curious fact that the sum of any quantity of consecutive odd numbers, beginning always with 1, is the square of that number. Thus:--

1 + 3 + 5 = 9 = 3 × 3.
1 + 3 + 5, etc., up to 17 = 81 = 9 × 9.
1 + 3 + 5, etc., up to 99 = 2500 = 50 × 50.

No. LV.--CHESS CAMEO

BY FRANK HEALEY

BLACK
+---+---+---+---+---+---+---+---+
| |.K.| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|.n.| |...| |.p.| |...| |
+---+---+---+---+---+---+---+---+
| |.N.| |.k.| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |.N.| |...| |...| |
+---+---+---+---+---+---+---+---+
| |.P.| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |.n.| Q |.P.| |
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE

White to play, and mate in three moves.

THE VERSATILE NUMBER

In the number 142857, if the digits which belong to it are in succession transposed from the first place to the end, the result is in each case a multiple of the original number. Thus:--

285714 = 142857 × 2
428571 = 142857 × 3
571428 = 142857 × 4
714285 = 142857 × 5
857142 = 142857 × 6

No. LVI.--CHESS CAMEO

BY J. E. CAMPBELL

BLACK
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| |.p.| |...|
+---+---+---+---+---+---+---+---+
|...| |...| p |...| p |...| B |
+---+---+---+---+---+---+---+---+
| |...| |...| |.k.| |...|
+---+---+---+---+---+---+---+---+
|...| |...| K |...| N |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |.P.|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |.R.| |
+---+---+---+---+---+---+---+---+
WHITE

White to play, and mate in three moves.

A PARADOX

By the following simple method, a plausible attempt is made to prove that 1 is equal to 2:--

Suppose that _a_ = _b_, then

_ab_ = _a_²

∴ _ab_ - _b_² = _a_² - _b_²

∴ _b_(_a_ - _b_) = (_a_ + _b_)(_a_ - _b_)

∴ _b_ = _a_ + _b_

∴ _b_ = 2_b_

∴ 1 = 2

This process only proves in reality that 0 × 1 = 0 × 2, which is true.

No. LVII.--CHESS CAMEO

_Double First Prize_

BY A. CYRIL PEARSON

BLACK
+---+---+---+---+---+---+---+---+
| |...| b |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| p |.K.| |...| p |
+---+---+---+---+---+---+---+---+
| |...| p |...| |.p.| |...|
+---+---+---+---+---+---+---+---+
|...| |...| B |.k.| |...| |
+---+---+---+---+---+---+---+---+
| |.b.| |...| |...| P |...|
+---+---+---+---+---+---+---+---+
|.N.| |.N.| |.P.| |...| |
+---+---+---+---+---+---+---+---+
| |.R.| |.P.| |.R.| |...|
+---+---+---+---+---+---+---+---+
|.b.| q |...| |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE

White to play, and mate in four moves.

QUICK CALCULATION

Few people know a very singular but simple method of calculating rapidly how much any given number of pence a day amounts to in a year. The rule is this:--Set down the given number of pence as pounds; under this place its half, and under that the result of the number of original pence multiplied always by five. Take, for example, 7d a day:--

£7 0 0
3 10 0
2 11
---------
£10 12 11
---------

The reason for this is evident as soon as we remember that the 365 days of a year may be split up into 240, 120, and 5, and that 240 happens to be the number of pence in a pound.

SCIENCE AT PLAY

No. LVIII.--THE GEARED WHEELS

A small wheel with ten teeth is geared into a large fixed wheel which has forty teeth. This small wheel, with an arrow mark on its highest cog, is revolved completely round the large wheel. How often during its course is the arrow pointing directly upwards? Here is a diagram of the starting position.

No. LIX.--ADVANCING BACKWARDS

Here is a most curious and interesting question:--When an engine is drawing a train at full speed from York to London, what part of the train at any given moment is moving _towards York_?

At any time, when the engine is drawing a train at full speed from York to London, that part of the flange of each wheel which is for the moment at its lowest is actually _moving backwards towards York_.

For any point, such as A, on the circumference of the tyre, describes in running along a series of curves, as shown by _full_ lines in the diagram; and any point, B, on the outer edge of the flange, follows a path shown by the _dotted_ curves.

If these lines are followed round with a pencil in the direction of the arrows, it will be found that the point on the flange actually moves _backwards_ as it passes _below the track_, while the point A, as it completes each curve, is _at rest_ for the instant on the track, just before it starts afresh. The speed of the train does not affect these very curious facts.

No. LX.--THE FIFTEEN BRIDGES

In the subjoined diagram A and B represent two islands, round which a river runs as is indicated, with fifteen connecting bridges, that lead from the islands to the river’s banks.

Can you contrive to pass in turn over all these bridges without ever passing over the same one twice?

ARRANGING THE DIGITS

In a school where two boys were taught to think out the bearings of their work, a sharp pupil remarked that 100 is represented on paper by the smallest digit and two cyphers, which are in themselves symbols of nothing. The master, quick to catch any signs of mental activity, took the opportunity to propound to his class the following ingenious puzzle:--How can the sum of 100 be represented exactly in figures and signs by making use of all the nine digits in their reverse order? This is how it is done:--

9 × 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 100.

Another ingenious method of using the nine digits, so that by simple addition they sum up to exactly 100, and each is used once only, is this:--

15 + 36 + 47 = 98 + 2 = 100

Here is another arrangement by which the nine digits written in their inverse order can be made to represent exactly 100:--

98 - 76 + 54 + 3 + 21 = 100.

Here is yet another way of arriving at 100 by using each of the digits, this time with an 0:--

40¹⁄₂
59³⁸⁄₇₆
--------
100

No. LXI.--LOOPING THE LOOP

Here is quite a pretty scientific experiment, which any one of a handy turn can construct and arrange:--

The spiral track is formed of two wires bent, and connected by curved cross-pieces. The upper twist is turned so that the ball starts on a horizontal course.

During the accelerated descent the ball acquires momentum enough to keep it on the vertical track, held outwardly against the wires by centrifugal force.

Convenient proportions are: height of spiral two feet, diameter six inches, and wire rails three-quarters of an inch apart.

No. LXII.--A MECHANICAL BIRD

A close approach to an ideal flying machine can be made with a little ingenuity. Two Y-shaped standards, secured to the backbone rod, support two wires which carry wings of thin silk, provided with light stays, and connected at their inner corners with the backbone by threads.

Rubber bands are attached to a loop on the inner end of the crank shaft, and secured to a post at the rear. These are twisted by turning the shaft with the cross wire, and when the tension is released the wings beat the air and carry the bird forward. It is known as Penaud’s mechanical bird, and has been sold as an attractive toy.

No. LXIII.--LINE OF SWIFTEST DESCENT

A simple apparatus constructed on the lines of this illustration will give an interesting proof of the laws which govern falling bodies on an inclined plane or on a curved path.

In the case of the inclined plane the ball is governed by the usual law which controls falling bodies. In that of the concave circular curve, as it is accelerated rapidly at the start, it makes its longer journey in quicker time. In the case of the cycloidal curve it acquires a high velocity. This curve has therefore been called “the curve of swiftest descent,” as a falling body passes over it in less time than upon any path except the vertical.

No. LXIV.--A CENTRIFUGAL RAILWAY

Here is another very simple and pretty illustration of the natural forces which come into play in “looping the loop.”

This scientific toy on a small scale may be easily made, if care is taken that the height of the higher end of the rails is to the height of the circular part in a greater ratio than 5 to 4.

A ball started at the higher end follows the track throughout, and at one point is held by centrifugal force against the under side of the rails, against the force of gravity.

No. LXV.--A QUESTION OF GRAVITY

If a ball is fired point blank from a perfectly horizontal gun, and travels half a mile over a level plain before it touches ground, and another similar ball is at the same moment dropped from the same height by some mechanical means, the two balls will touch ground simultaneously. The flight, however long, of one through the air has no influence upon the force of gravity, which draws it earthward at the same resistless rate as it draws the other that is merely dropped.

A SHORT CUT

A quick method of multiplying any number of figures by 5 is to divide them by 2, annexing a cypher to the result when there is no remainder, and if there is any remainder annexing a 5. Thus:--

464 × 5 = 2320; 464 ÷ 2 = 232, annex 0, = 2320.
753 × 5 = 3765; 753 ÷ 2 = 376, annex 5, = 3765.

No. LXVI.--A DUCK HUNT

A duck begins to swim round the edge of a circular pond, and at the same moment a water spaniel starts from the middle of the pond in pursuit of it.

If both swim at the same pace, how must the dog steer his course so that he is sure in any case to overtake the duck speedily?

THE MAGIC OF DATES

LOUIS NAPOLEON, EMPEROR 1852

1852 1852 1852
date 1 date 1 date 1
of 8 of 8 of 8
his 0 Empress’s 2 their 5
birth 8 birth 6 marriage 3
---- ---- ----
1869 1869 1869

Thus, by a most remarkable series of coincidences, the principal dates of the Emperor and Empress of the French added, as is shown above, to the year of the Emperor’s accession, express in each instance the year before his fall.

No. LXVII.--GEOMETRY WITH DOMINOES

In this domino diagram we have a pretty and practical proof that the squares of the sides containing the right angle in any right-angled triangle are together equal to the square of the side opposite to the right angle.

Each stone forms two squares, and it is easily seen that the number of squares which make up the whole square on the line opposite the right angle are equal to the number of those which make up the two whole squares on the lines which contain that angle.

A second point to be noticed is that the number of pips on the large square are equal to the number on the other two squares combined, an arrangement of the stones which forms quite a game of patience to reproduce, if this pattern is not at hand.

No. LXVIII.--TO COLOUR MAPS

Four colours at most are needed to distinguish the surfaces of separate districts on any plane map, so that no two with a common boundary are tinted alike.

On this diagram A, B, and C, are adjoining districts, on a plane surface, and X borders, in one way or another, upon each.

It is clearly impossible to introduce a fifth area which shall so adjoin these four districts as to need another tint.

A FREAK OF FIGURES

Here is another freak of figures:--

9 × 1 - 1 = 8
9 × 21 - 1 = 188
9 × 321 - 1 = 2888
9 × 4321 - 1 = 38888
9 × 54321 - 1 = 488888
9 × 654321 - 1 = 5888888
9 × 7654321 - 1 = 68888888
9 × 87654321 - 1 = 788888888
9 × 987654321 - 1 = 8888888888

No. LXIX.--THE TETHERED BIRD

A bird made fast to a pole six inches in diameter by a cord fifty feet long, in its flight first uncoils the cord, _keeping it always taut_, and then recoils it in the reverse direction, rewinding the coils close together. If it starts with the cord fully coiled, and continues its flight until it brings up against the pole, how far does it fly in its double course?

STRIKE IT OUT

Ask a person to write down in a line any number of figures, then to add them all together as units, and to subtract the result from the sum set down. Let him then strike out any one figure, and add the others together as units, telling you the result.

If this has been correctly done, the figure struck out can always be determined by deducting the final total from the multiple of 9 next above it. If the total happens to be a multiple of 9, then a 9 was struck out.

No. LXX.--THE MOVING DISC AND THE FLY

A fly, starting from the point A, just outside a revolving disc, and always making straight for its mate at the point B, crosses the disc in four minutes, while the disc is revolving twice. What effect has the revolution of the disc on the path of the fly?

A MAGIC SQUARE

This Magic Square is so arranged that the product of the continued multiplication of the numbers in each row, column, or diagonal is 4096, which is the cube of the central 16.

+---+---+---+
| 8|256| 2|
+---+---+---+
| 4| 16| 64|
+---+---+---+
|128| 1| 32|
+---+---+---+

No. LXXI.--A SHUNTING PUZZLE

The railway, D E F, has two sidings, D B A and F C A, connected at A. The rails at A, common to both, are long enough to hold a single wagon such as P or Q, but too short to admit the whole of the engine R, which, if it runs up either siding, must return the same way.

How can the engine R be used to interchange the wagons P and Q without allowing any flying shunts?--From _Ball’s Mathematical Recreations_.

No. LXXII.--A CURIOUS FACT

It is a little known and very interesting fact that an equilateral triangle can easily be drawn by rule of thumb in the following way:--Take a triangle of any shape or size, and on each of its sides erect an equilateral triangle. Find and join the centres of these, and a fourth equilateral triangle is always thus formed, as shown by the dotted lines.

These centres are _centres of gravity_, and they are symmetrically distributed around the centre of gravity of the original triangle.

The figure formed by joining them must therefore be symmetrical, and, as in this case, it is a triangle, it _must be_ always equilateral.

No. LXXIII.--TRY THIS EXPERIMENT

There can be no better instance of how the eye may be deceived than is so strikingly afforded in these very curious diagrams:--

The square which obviously contains sixty-four small squares, is to be cut into four parts, as is shown by the thicker lines. When these four pieces are quite simply put together, as shown in the second figure, there seem to be sixty-five squares instead of sixty-four.

This phenomena is due to the fact that the edges of the four pieces, which lie along the diagonal A B, do not exactly coincide in direction. In reality they _include a very narrow diamond_, not easily detected, whose area is just equal to that of one of the sixty-four small squares.

FIGURES IN SWARMS

Very curious are the results when the nine digits in reverse order are multiplied by 9 and its multiples up to 81. Thus:--

987654321 × 9 = 8888888889
× 18 = 17777777778
× 27 = 26666666997
× 36 = 35555555556
× 45 = 44444444445
× 54 = 53333333334
× 63 = 62222222223
× 72 = 71111111112
× 81 = 80000000001

It will be seen that the figures by which the reversed digits are multiplied reappear at the beginning and end of each result except the first, and that the figures repeated between them are to be found by dividing the divisors by 9 and subtracting the result from 9. Thus, 54 ÷ 9 = 6, and 9 - 6 = 3.

No. LXXIV.--A TRIANGLE OF TRIANGLES

In this nest of triangles there are no less than six hundred and fifty-three distinct triangles of various shapes and sizes.

No. LXXV.--PHARAOH’S SEAL

In a chamber of the Great Pyramid an ancient Egyptian jar was found, marked with the device now known as Pharaoh’s seal.

Can you count the number of triangles or pyramids, of many sizes, but all of similar shape that are expressed on it? Solvers should draw the figure on a larger scale.

MAKING CUBES

It is interesting to note that the repeated addition of odd numbers to one another can be so arranged as to produce cube numbers in due sequence. Thus:--

1 = 1 × 1 × 1
3 + 5 = 2 × 2 × 2
7 + 9 + 11 = 3 × 3 × 3
13 + 15 + 17 + 19 = 4 × 4 × 4
21 + 23 + 25 + 27 + 29 = 5 × 5 × 5

and so on, to any extent.

No. LXXVI.--ROUND THE GARDEN

In a large old-fashioned garden walks were arranged round a central fountain in the shape of a Maltese cross.

If four persons started at noon from the fountain, walking round the four paths at two, three, four and five miles an hour respectively, at what time would they meet for the third time at their starting-point, if the distance on each track was one-third of a mile?

A NICE SHORT CUT

When the tens of two numbers are the same, and their units added together make ten, multiply the units together, increase one of the tens by unity, and multiply it by the other ten. The result is the product of the two original numbers, if the first result follows the other. Thus:--

43 × 47 = 2021.

No. LXXVII.--A JOINER’S PUZZLE

Can you cut Fig. A into two parts, and so rearrange these that they form either Fig. B or Fig. C?

+--------------+
| |
+--------------+ | +--+
| | | |
| | +-----------------+ | |
| | | | | |
| | | | | |
| | | | | |
| | | | | |
| | | | | |
| | | | | |
| | | | | |
| | | | +--+ |
| | | | | |
+--------------+ +-----------------+ +--------------+
A B C

The two parts of A must not be _turned round_ to form B or C, but must retain their original direction.

A CALCULATION

Coal may fail us, but we can never run short of material for “words that burn.” It has been calculated that if a man could read 100,000 words in an hour, and there were 4,650,000 men available, they could not pronounce the possible variations which could be formed from the alphabet in 70,000 years!

A PARADOX

It is possible, in a sense, by the following neat method, to take 45 from 45, and find that 45 remains:--

987654321 = 45.
123456789 = 45.
---------
864197532 = 45.

No. LXXVIII.--THE BROKEN OCTAGON

Cut out in stiff cardboard four pieces shaped as Fig. 1, four as Fig. 2, and four as Fig. 3, taking care that they are all exactly true to pattern in shape and proportion to one another.

Now see whether you can put the twelve pieces together so as to form a perfect octagon.

PROPERTIES OF SEVEN

Here is a proof that 7, if it cannot rival the mystic 9, has quaint properties of its own:--

15873 × 7 = 111111
31746 × 7 = 222222
47619 × 7 = 333333
63492 × 7 = 444444
79365 × 7 = 555555
95238 × 7 = 666666
111111 × 7 = 777777
126984 × 7 = 888888
142857 × 7 = 999999

A SWARM OF EIGHTS

Here is an arithmetical curiosity:--

9 × 9 + 7 = 88
9 × 98 + 6 = 888
9 × 987 + 5 = 8888
9 × 9876 + 4 = 88888
9 × 98765 + 3 = 888888
9 × 987654 + 2 = 8888888
9 × 9876543 + 1 = 88888888
9 × 98765432 + 0 = 888888888

No. LXXIX.--AT A DUCK POND

A farmer’s wife kept a pure strain of Aylesbury ducks for market on a square pond, with a duck-house at each corner. As trade grew brisk she found that she must enlarge her pond. An ingenious neighbour undertook to arrange this without altering the shape of the pond, and without disturbing the duck-houses. What was his plan?

○-------------○
| |
| |
| |
| |
| |
| |
○-------------○

STRANGE SUBTRACTION

It would seem impossible to subtract 69 from 55, but it can be arranged thus, with six as a remainder:--

SIX IX XL
IX X L
-------------
S I X
=============

No. LXXX.--ALL ON THE SQUARE

Cut out in cardboard twenty triangular pieces exactly the size and shape of this one, and try to place them together so that they form a perfect square.

ANOTHER MYSTIC NUMBER

The decimal equivalent of ¹⁄₁₃ is .076923. This (omitting the point), multiplied by 1, 3, 4, 9, 10, or 12, yields results in which the same figures appear in varied order, but similar sequence, and multiplied by 2, 5, 6, 7, 8, or 11, it yields a different series, with similar characteristics. Thus:--

76923 × 1 = 76923
× 3 = 230769
× 4 = 307692
× 9 = 692307
× 10 = 769230
× 12 = 923076

76923 × 2 = 153846
× 5 = 384615
× 6 = 461538
× 7 = 538461
× 8 = 615384
× 11 = 846153

DON’T BUY IT TO TRY IT

A kaleidoscope cylinder contains twenty small pieces of coloured glass. As we turn it round, or shake it, so as to make ten changes of pattern every minute, it will take the inconceivable space of time of 462,880,899,576 years and 360 days to exhaust all the possible symmetrical variations. (The 360 days is good!)

No. LXXXI.--PINS AND DOTS

Here is an amusing little exercise for the ingenuity of our solvers.

*-----------*-----------*-----------*-----------*-----------*
| \ / | \ / | \ / | \ / | \ / |
| \ / | \ / | \ / | \ / | \ / |
| * | * | * | * | * |
| / \ | / \ | / \ | / \ | / \ |
| / \ | / \ | / \ | / \ | / \ |
*-----------*-----------*-----------*-----------*-----------*
| \ / | \ / | \ / | \ / | \ / |
| \ / | \ / | \ / | \ / | \ / |
| * | * | * | * | * |
| / \ | / \ | / \ | / \ | / \ |
| / \ | / \ | / \ | / \ | / \ |
*-----------*-----------*-----------*-----------*-----------*
| \ / | \ / | \ / | \ / | \ / |
| \ / | \ / | \ / | \ / | \ / |
| * | * | * | * | * |
| / \ | / \ | / \ | / \ | / \ |
| / \ | / \ | / \ | / \ | / \ |
*-----------*-----------*-----------*-----------*-----------*
| \ / | \ / | \ / | \ / | \ / |
| \ / | \ / | \ / | \ / | \ / |
| * | * | * | * | * |
| / \ | / \ | / \ | / \ | / \ |
| / \ | / \ | / \ | / \ | / \ |
*-----------*-----------*-----------*-----------*-----------*
| \ / | \ / | \ / | \ / | \ / |
| \ / | \ / | \ / | \ / | \ / |
| * | * | * | * | * |
| / \ | / \ | / \ | / \ | / \ |
| / \ | / \ | / \ | / \ | / \ |
*-----------*-----------*-----------*-----------*-----------*

Take six sharp pins, and puzzle out how to stick them into six of the black dots, so that no two pins, are on the same line, in any direction, vertical, horizontal, or diagonal.

No. LXXXII.--A TRICKY COURSE

The middle of a large playground was paved with sixty-four square flagstones of equal size, which are numbered on this diagram from one to sixty-four.

+----+----+----+----+----+----+----+----+
| _1_| _9_|_17_|_25_|_33_|_41_|_49_|_57_|
+----+----+----+----+----+----+----+----+
| _2_|_10_|_18_|_26_|_34_|_42_|_50_|_58_|
+----+----+----+----+----+----+----+----+
| _3_|_11_|_19_|_27_|_35_|_43_|_51_|_59_|
+----+----+----+----+----+----+----+----+
| _4_|_12_|_20_|_28_|_36_|_44_|_52_|_60_|
+----+----+----+----+----+----+----+----+
| _5_|_13_|_21_|_29_|_37_|_45_|_53_|_61_|
+----+----+----+----+----+----+----+----+
| _6_|_14_|_22_|_30_|_38_|_46_|_54_|_62_|
+----+----+----+----+----+----+----+----+
| _7_|_15_|_23_|_31_|_39_|_47_|_55_|_63_|
+----+----+----+----+----+----+----+----+
| _8_|_16_|_24_|_32_|_40_|_48_|_56_|_64_|
+----+----+----+----+----+----+----+----+

One of the schoolmasters, who had a head for puzzles, took his stand upon the square here numbered 19, and offered a prize to any boy who, starting from the square numbered 46, could make his way to him, passing through every square once, and only once. It was after many vain attempts that the course was at last discovered. Can you work it out?

No. LXXXIII.--FOR THE CHILDREN

Place twelve draughtsmen, or buttons, in a square, so that you count four along each side of it, thus:--

● ● ● ●
● ●
● ●
● ● ● ●

Now take the same men or buttons, and arrange them so that they form another square, and you can count five along each side of it.

A GAME OF NINES

Here is a good specimen of the eccentricities and powers of numbers:--

153846 × 13 = 1999998
230769 × 13 = 2999997
307692 × 13 = 3999996
384615 × 13 = 4999995
461538 × 13 = 5999994
538461 × 13 = 6999993
615384 × 13 = 7999992
692307 × 13 = 8999991

No. LXXXIV.--TELL-TALE TABLES

She was quite an old maid, and her age was a most absolute secret. Determined to discover it, her scapegrace nephew, on Christmas Eve, produced these tables, and asked her with well simulated innocence on which of them she could see the number of her age.

+--+--+--+ +--+--+--+ +--+--+--+
| 1|23|45| | 2|23|46| |16|27|54|
+--+--+--+ +--+--+--+ +--+--+--+
| 3|25|47| | 3|26|47| |17|28|55|
+--+--+--+ +--+--+--+ +--+--+--+
| 5|27|49| | 6|27|50| |18|29|56|
+--+--+--+ +--+--+--+ +--+--+--+
| 7|29|51| | 7|30|51| |19|30|57|
+--+--+--+ +--+--+--+ +--+--+--+
| 9|31|53| |10|31|54| |20|31|58|
+--+--+--+ +--+--+--+ +--+--+--+
|11|33|55| |11|34|55| |21|48|59|
+--+--+--+ +--+--+--+ +--+--+--+
|13|35|57| |14|35|58| |22|49|60|
+--+--+--+ +--+--+--+ +--+--+--+
|15|37|59| |15|38|59| |23|50|61|
+--+--+--+ +--+--+--+ +--+--+--+
|17|39|61| |18|39|62| |24|51|62|
+--+--+--+ +--+--+--+ +--+--+--+
|19|41| | |19|42| | |25|52| |
+--+--+ A| +--+--+ B| +--+--+ C|
|21 43| | |22 43| | |26|53| |
+--+--+--+ +--+--+--+ +--+--+--+

+--+--+--+ +--+--+--+ +--+--+--+
| 8|27|46| | 4|23|46| |32|43|54|
+--+--+--+ +--+--+--+ +--+--+--+
| 9|28|47| | 5|28|47| |33|44|55|
+--+--+--+ +--+--+--+ +--+--+--+
|10|29|56| | 6|29|52| |34|45|56|
+--+--+--+ +--+--+--+ +--+--+--+
|11|30|57| | 7|30|53| |35|46|57|
+--+--+--+ +--+--+--+ +--+--+--+
|12|31|58| |12|31|54| |36|47|58|
+--+--+--+ +--+--+--+ +--+--+--+
|13|40|59| |13|36|55| |37|48|59|
+--+--+--+ +--+--+--+ +--+--+--+
|14|41|60| |14|37|60| |38|49|60|
+--+--+--+ +--+--+--+ +--+--+--+
|15|42|61| |15|38|61| |39|50|61|
+--+--+--+ +--+--+--+ +--+--+--+
|24|43|62| |20|39|62| |40|51|62|
+--+--+--+ +--+--+--+ +--+--+--+
|25|44| | |21|44| | |41|52| |
+--+--+ D| +--+--+ E| +--+--+ F|
|26|45| | |22|45| | |42|53| |
+--+--+--+ +--+--+--+ +--+--+--+

From her answer he was able to calculate that the old lady was fifty-five.

The tell-tale tables disclosed her age thus:--As it appeared in tables A, B, C, E, and F, he added together the numbers at the top left-hand corners, and found the total to be fifty-five. This rule applies in all cases.

No. LXXXV.--A PAPER PUZZLE

Of the many paper-cutting tricks which appeal to us none is more simple and attractive than this:--

+---+
| |
| |
+-----+ +-----+
| |
+-----+ +-----+
/ \ | | / \
\ / | | \ /
+-+ | | +-+
| | | | | |
| | | | | |
| | | | | |
| | +---+-+-+---+ | |
+-+ | | | +-+
+---+ | +-+-+ | +---+
| +---+---+-+-+---+---+ |
| | |
+-------------+-------------+

Take a piece of paper, say 5 inches by 3 inches, but any oblong shape and size will do, and after folding it four times cut it lengthways up the centre. Unfold the pieces, and to your surprise you will find a perfect cross and other pieces in pairs of the shapes shown above. The puzzle is how to fold the paper.

+--------------+----------+
|b d¦ |
| ¦ |
| ¦ |
| e¦ |
| ¦ |
| ¦ |
|a c¦ |
+--------------+----------+

The paper must be folded first so that B comes upon C, then so that A comes upon D, then from D to C, and lastly from E to C. If it is now cut lengthways exactly along the centre the figures shown on the original diagram will be formed, which resemble a cross and lighted candles on an altar.

No. LXXXVI.--A HOME-MADE PUZZLE

Take a thin board, about eight inches square, and mark it out into thirty-six equal parts; bore a hole in the centre of each part, and then fit in a small wooden peg, leaving about a quarter inch above the surface, as is shown in Fig. 1, the section below the diagram.

Prepare thirty-six pieces of white or coloured cardboard of the length A to B, and place them over the pegs in any direction in which they will fit so as to form some such symmetrical pattern as is given on the second diagram, putting two holes only on each peg. Chess-players will see that this is the regular knight’s move.

Quite a number of beautiful designs can be thus formed, and those who have not the means at hand for making a complete set can enjoy the puzzle by merely marking out thirty-six squares, and drawing lines from centre to centre of the exact length from A to B, with black or coloured pencils.

No. LXXXVII.--LOYD’S MITRE PROBLEM

Divide this figure into four similar and equal parts.

A PRETTY PROBLEM

The solution of the pretty little problem: place three twos in three different groups, so that twice the first group, or half the third group equals the second group, is this:--

2 1 2 2 + 2
----- = - 2 - - = 1 ----- = 2
2 + 2 2 2 2

No. LXXXVIII.--CONTINUOUS LINES

The following figure, which represents part of a brick wall, cannot be marked out along all the edges of the bricks in less than six continuous lines without going more than once over the same line:--

Here, in strong contrast to the simple figure given above, which could not be traced without lifting the pen six times from the paper, is an intricate design, the lines of which, on the upper or on the lower half, can be traced without any break at all.

The general rule that governs such cases is, that where an uneven number of lines meet a fresh start has to be made. In the diagram now given the only such points are at the extremities of the upper and lower halves of the figure at A and X. At all other points two, or four, or six lines converge, and there is no break of continuity in a tracing of the figure.

No. LXXXIX.--CUT OFF THE CORNERS

Can you suggest quite a simple and practical way to fix the points on the sides of a square which will be at the angles of an octagon formed by cutting off equal corners of the square, as shown below?

A E F B
+------+--------+------+
| / \ |
| / \ |
|/ \|
M+ +G
| |
| |
| |
L+ +H
|\ /|
| \ / |
| \ / |
+------+--------+------+
C K I D

MYSTIC FIGURES

Very interesting and curious are the properties of the figures 142857, used in varied order but always in similar sequence, in connection with 7 and 9:--

142857 × 7 = 999999 ÷ 9 = 111111
285714 × 7 = 1999998 ÷ 9 = 222222
428571 × 7 = 2999997 ÷ 9 = 333333
571428 × 7 = 3999996 ÷ 9 = 444444
714285 × 7 = 4999995 ÷ 9 = 555555
857142 × 7 = 5999994 ÷ 9 = 666666

No. XC.--THE FIVE TRIANGLES

The subjoined diagram shows how a square with sides that measure each 12 yards can be divided into five triangles, no two of which are of equal area, and of which the sides and areas can be expressed in yards by whole numbers:--

The areas of these triangles are 6, 12, 24, 48, and 54 square yards respectively, and the sum of these, 144 square yards, is the area of the square.

CURIOUS COINCIDENCES

Our readers may remember the remarkable fact that the figures of the sum, £12, 12s. 8d., when written thus, 12,128, exactly represent the number of farthings it contains. Now this, so far as we know, is the only instance of the peculiarity, but there are at least five other cases which come curiously near to it. They are these:--

£ s. d.
9 9 6 = 9096 farthings
6 6 4 = 6064 „
3 3 2 = 3032 „
10 10 6¹⁄₂ = 10106 „
13 13 8¹⁄₂ = 13138 „

No. XCI.--PLACING A LADDER

If a ladder, with rungs 1 foot apart, rests against a wall, and its thirteenth rung is 12 feet above the ground, the foot of the ladder is 25 feet from the wall.

_Proof._--Drop a perpendicular from A to B. Then, as A B C is a right angle, and the squares on A C, A B, are 169 feet and 144 feet, the square on C B must be 25 feet, and the length of C B is 5 feet. We thus move 5 feet towards the wall in going 13 feet up the ladder, and in mounting 65 feet (five times as far) we must cover 25 feet.

No. XCII.--GRACEFUL CURVES

A prettily ingenious method of dividing the area of a circle into quarters, each of them a perfect curve, with perimeter (or enclosing line) equal to the circumference of the circle, and with which four circles can be formed, is clearly shown by the subjoined diagrams:--

NIGHTS AT A ROUND TABLE

The host of a large hotel at Cairo noticed that his Visitors’ Book contained the names of an Austrian, a Brazilian, a Chinaman, a Dane, an Englishman, a Frenchman, a German, and a Hungarian. Moved by this curious alphabetical list, he offered them all free quarters and the best of everything if they could arrange themselves at a round dining-table so that not one of them should have the same two neighbours on any two occasions for 21 successive days.

The following is one of many ways in which this arrangement can be made, and it seems to be the simplest of them all.

Number the persons 1 to 8; and for our first day set them down in numerical order _except that the two centre ones (4 and 5) change places_:

(1st day)--1 2 3 5 4 6 7 8

Keep the 1 and the 7 unaltered but double each of the other numbers. When the product is greater than 8, divide by 7, and only set down the remainder. Thus we get:

(8th day)--1 4 6 3 8 5 7 2

(Here the fourth figure 3 is 5 × 2 ÷ 7, giving _remainder_ 3, and so on.)

Repeat this operation once more:

(15th day)--1 8 5 6 2 3 7 4

To fill in the intermediate days we have only to keep 1 unchanged and let the remaining numbers run downwards in _simple numerical order_, following 8 with 2, 2 with 3, and so on. Thus:--

1st day--1 2 3 5 4 6 7 8
2nd day--1 3 4 6 5 7 8 2
3rd day--1 4 5 7 6 8 2 3
4th day--1 5 6 8 7 2 3 4
5th day--1 6 7 2 8 3 4 5
6th day--1 7 8 3 2 4 5 6
7th day--1 8 2 4 3 5 6 7
-------------------------
8th day--1 4 6 3 8 5 7 2
9th day--1 5 7 4 2 6 8 3
10th day--1 6 8 5 3 7 2 4
11th day--1 7 2 6 4 8 3 5
12th day--1 8 3 7 5 2 4 6
13th day--1 2 4 8 6 3 5 7
14th day--1 3 5 2 7 4 6 8
-------------------------
15th day--1 8 5 6 2 3 7 4
16th day--1 2 6 7 3 4 8 5
17th day--1 3 7 8 4 5 2 6
18th day--1 4 8 2 5 6 3 7
19th day--1 5 2 3 6 7 4 8
20th day--1 6 3 4 7 8 5 2
21st day--1 7 4 5 8 2 6 3

This completes the schedule. It will be found on examination that every number is between every pair of the other numbers once, and once only.

In order to reduce our first-day ring to exact numerical order we have only to interchange the numbers 4 and 5 throughout. The first three lines for example would then become:

1 2 3 4 5 6 7 8
1 3 5 6 4 7 8 2
1 5 4 7 6 8 2 3, etc.

or, by putting letters for figures,

A B C D E F G H
A C E F D G H B
A E D G F H B C, etc.

An arrangement of the guests is thus arrived at for twenty-one successive days, so that not one of them has the same two neighbours on any two occasions.

No. XCIII.--MAKING MANY SQUARES

Can you apply the two oblongs drawn below to the two concentric squares, so as to produce thirty-one perfect squares?

+------+ +------+
| | | |
| | | |
+----------------------------------+ +------+ +------+
| | | | | |
| | | | | |
| +--------------------+ | +------+ +------+
| | | | | | | |
| | | | | | | |
| | | | +------+ +------+
| | | | | | | |
| | | | | | | |
| | | | +------+ +------+
| | | | | | | |
| | | | | | | |
| +--------------------+ | +------+ +------+
| | | | | |
| | | | | |
+----------------------------------+ +------+ +------+
| | | |
| | | |
+------+ +------+

No. XCIV.--CUT ACROSS

Take a piece of cardboard in the form of a Greek cross with arms, as shown here, and divide it by two straight cuts, so that the pieces when reunited form a perfect square.