Black and White

By Sam Loyd. White to mate in two moves.

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1. Qa8! and the queen or the rook must mate. From E.B. Cook et al., American Chess-Nuts, 1868.

December 24, 2011January 30, 2013 | Puzzles

What Are We?

A riddle by Jonathan Swift:

We are little airy creatures,
All of different voice and features:
One of us in glass is set,
One of us you’ll find in jet,
T’other you may see in tin,
And the fourth a box within;
If the fifth you should pursue,
It can never fly from you.

What are we?

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Vowels.

December 22, 2011January 30, 2013 | Puzzles

Growing Pains

A problem from the Soviet Mathematical Olympiad:

Two hundred students are arranged in 10 rows of 20 children. The shortest student in each column is identified, and the tallest of these is marked A. The tallest student in each row is identified, and the shortest of these is marked B. If A and B are different people, which is taller?

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B is taller: If A and B stand in the same row, then B is taller, since we know that B is the tallest student in his row. If A and B stand in the same column, then again B is taller, since we know that A is the shortest student in his column. If A and B share neither a row nor a column, then let C be the student who’s in the same column as A and the same row as B. Then C is shorter than B (who is the tallest in his row) and taller than A (who is the shortest in his column), so B > C > A. In every case, then, B is taller than A.

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December 20, 2011January 30, 2013 | Puzzles

Black and White

loquin chess problem

By Paul Loquin. White to mate in two moves.

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1. Nb7, followed by 2. Qe4#. From Le Palamède, via Charles Tomlinson, Amusements in Chess, 1845.

December 18, 2011January 30, 2013 | Puzzles

Odd and Even

Put the integers 1, 2, 3, … n in any order and call them a₁, a₂, a₃, … a_n. Then form the product

P = (a₁ – 1) × (a₂ – 2) × (a₃ – 3) … × (a_n – n).

Now: If n is odd, prove that P is even.

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If n is odd then the sequence (1, 2, 3, … n) begins and ends with an odd number. This means it contains one more odd number than even. Likewise (a₁, a₂, … a_n), which are only the same numbers in a different order. This means that two odd numbers must end up in the same bracket — there are not enough even numbers available to pair each with an odd. And if two odd numbers appear in the same bracket, they’ll produce an even difference and hence an even product. From Ross Honsberger, Ingenuity in Mathematics, 1970.

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December 17, 2011January 30, 2013 | Puzzles

All in the Family

Two men are brothers-in-law if one is married to the other’s sister. What is the largest possible group of men in which each is brother-in-law to each of the others?

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Three. Three men can be mutual brothers-in-law if A marries B’s sister, B marries C’s sister, and C marries A’s sister. But D has no way to join this group. He cannot ask his sister to marry A, B, or C, because they are already spoken for. And if he himself marries the sister of, say, A, then he cannot be a brother-in-law to C unless C or D is A’s brother, which implies a violation of the law against consanguinity. (If a man may marry his own sister, then there’s no limit to the number of mutual brothers-in-law.) (David L. Silverman, “A Problem of Relations,” Journal of Recreational Mathematics 3:3, July 1970)

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December 15, 2011January 30, 2013 | Puzzles

Sweet and Sour

ant locus puzzle

An ant will always position itself so that it’s precisely twice as far from vinegar as from honey. If we put a dab of vinegar at A and a dab of honey at B and we release a troop of ants, what formation will they take up?

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They’ll form a circle:

December 11, 2011January 30, 2013 | Puzzles

Black and White

fink chess problem

A.J. Fink, Good Companions, February 1917. White to mate in two moves.

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1. Qd2 and Black can’t avoid a fatal discovered check.

December 10, 2011January 30, 2013 | Puzzles

“Paradox”

From Miscellanea Curiosa: or, Entertainments for the Ingenious of Both Sexes, January 1734:

One evening, as I walk’d to take the Air,
I chanc’d to overtake two Ladies fair;
Each by the Hand a lovely Boy did lead,
To whom in courteous Manner thus I said:
Ladies! so far oblige me as to shew
How near akin these Boys are unto you?
They, smiling, quickly made this dark Reply,
Sons to our Sons they are, we can’t deny:
Though it seem strange, they are our Husbands’ Brothers,
And likewise each is Uncle to the other:
They both begot, and born in Wedlock were,
And we their Mothers and Grandmothers are.
Now try if you this Mystery can declare.

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Two widow Ladies married were, Each to the other’s Son; And they both pregnant did appear E’er one full Year was run. The Consequence of which did prove, To each a charming Boy; This did cement their Husband’s Love, And added to their Joy. By this Event likewise it’s plain They did commence Grandmothers, And that their Husbands did obtain Two young delightful Brothers. A Brother you may justly call Each, to the other’s Father: Uncles they were reciprocal, You easily may gather.

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December 8, 2011January 30, 2013 | Puzzles

Countdown

Will a prime number ever appear in this series?

9
98
987
9876
98765
987654
9876543
98765432
987654321
9876543210
98765432109
987654321098
9876543210987
…

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No. Eliminate the even numbers and those ending in 5. The sum of the digits in each of the remaining numbers will always produce a multiple of 3, so the numbers themselves will always be divisible by 3.

December 6, 2011January 30, 2013 | Puzzles