Black and White

shinkman chess problem

By William Anthony Shinkman, 1880. White to mate in two moves.

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1. Rd1! and the queen will mate.

August 11, 2017August 11, 2017 | Puzzles

The Last Detail

A puzzle by R.P. Cross:

Find the last digit in the evaluation of $\sum_{n = 1}^{100}n!$

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It would take a lot of pencils to work this out by hand, but an insight makes it easy: For any n ≥ 5, n! must have the factors 2 and 5 and hence be a multiple of 10, with zero as a final digit. So to solve the puzzle we only have to work out 1! + 2! + 3! + 4! = 33. The last digit is 3. (“A Property of Large Factorial Numbers,” Mathematical Gazette 81:490 [March 1997], 80-81.)

August 7, 2017 | Puzzles · Science & Math

Black and White

kalendovský chess problem

By Jan Kalendovský. White to mate in two moves.

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1. Rf2! Bxf2+ 2. Kxf2#

July 24, 2017July 24, 2017 | Puzzles

Late Again

kendall/thomas railroad puzzle

A problem from P.M.H. Kendall and G.M. Thomas’ Mathematical Puzzles for the Connoisseur, 1962: A road runs parallel to a railway until it bends to cross it, as shown. A man normally cycles to work along the road at a constant speed of 12 mph, and when he reaches the crossing he’s normally overtaken by a train traveling in the same direction. One day he was 25 minutes late for work and found that the train passed him 6 miles before the crossing. What was the speed of the train?

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Suppose the man normally reaches the crossing, B, at 8 a.m. Then on a typical day he’d reach A, 6 miles before the crossing, at 7:30 a.m. When the man is 25 minutes late, he’ll reach B at 8:25 a.m. and A at 7:55 a.m. Hence the train takes 5 minutes to travel the 6 miles from A to B. Therefore its speed is 72 mph.

July 15, 2017July 14, 2017 | Puzzles

Cheryl’s Birthday

This question appeared in the 2015 Singapore and Asian Schools Math Olympiad, a competition for 14-year-old students from Singapore, Thailand, Vietnam, China, and the U.K. (I’ve amended the language a bit):

Albert and Bernard have just become friends with Cheryl, and they want to know when her birthday is. Cheryl gives them a list of 10 possible dates:

May		15	16			19
June				17	18
July	14		16
August	14	15		17

Cheryl then tells Albert and Bernard separately the month and the day of her birthday, respectively.

Albert: I don’t know when Cheryl’s birthday is, but I know that Bernard doesn’t know it either.

Bernard: At first I didn’t know when Cheryl’s birthday is, but I know now.

Albert: Then I also know when Cheryl’s birthday is.

When is Cheryl’s birthday?

Singapore TV presenter Kenneth Kong posted the question online, and it went viral in a matter of days. The competition organizers had intended it to “sift out the better students” and expected that 40 percent of the competitors would find the solution. What is it?

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Cheryl’s birthday is July 16. Albert’s first statement is unsurprising — he knows only the month of Cheryl’s birthday, and each month on the list includes several possible days. But the fact that he knows that Bernard doesn’t know it is important — it tells us that the month can’t be May or June, because both of those months contain unambiguous dates. For example, if Cheryl’s birthday were May 19, then Bernard would know immediately that she was born in May, as that’s the only month whose 19th day appears on the list. Similarly, if Bernard knew the day of Cheryl’s birthday was the 18th, he’d know immediately that she was born in June. The fact that Albert can exclude these possibilities shows that he knows she wasn’t born in May or June; she must have been born in July or August. Bernard’s statement shows that he’s followed this and now knows that the month must be either July or August. And the fact that he now knows the full birthday shows that Cheryl was born on the 15th, 16th, or 17th — if she’d been born on the 14th then he’d still be uncertain, as both July 14 and August 14 fall on the list. Albert has narrowed the list of possibilities to July 16, August 15, and August 17. As he now declares that he knows the birthday unambiguously, it must be July 16 — if it fell in August he’d have no way to distinguish between August 15 and August 17.

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July 12, 2017 | Puzzles

Patrolling the Palace

A puzzle by James Tanton:

King Tricho lives in a palace in which every room is a triangle:

patrolling the palace

Before retiring for the night he’d like to inspect it. Is there a path that will let him visit each room once and only once? He can start anywhere.

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No, there isn’t. Color the cells as shown: Any path that the king takes must alternate between dark and light rooms, so the king can succeed only if there are an equal number of each (or if their numbers differ by 1). In the diagram there are 12 dark rooms and 10 light rooms, so there’s no way to visit all of them on one connected tour, unless the path travels outside the palace. (From Tanton’s “A Dozen Questions About a Triangle,” Math Horizons 9:4 [April 2002], 23-28.)

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July 11, 2017July 11, 2017 | Puzzles

Black and White

vorrath chess problem

A pretty problem by August Vorrath. White to mate in two moves.

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1. Rd8! cxd5 2. Rc8# From Brownson’s Chess Journal, May 1877.

July 4, 2017July 4, 2017 | Puzzles

Spirits of the Departed

A wine merchant has three sons. When he dies, he leaves them seven barrels that are full of wine, seven that are half-full, and seven that are empty. His will requires that each son receive the same number of full, half-full, and empty barrels. Can this be done?

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Yes. Combine four of the half-full barrels to make two full ones. Now each son can receive three full, one half-full, and three empty barrels. 06/28/2017 A number of readers pointed out a simpler solution: Just pour half of a full barrel into an empty barrel! I should have seen this. The puzzle appears in James F. Fixx’s More Games for the Superintelligent; he doesn’t give the original source.

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June 27, 2017June 28, 2017 | Puzzles

Prime Magic

berloquin prime puzzle

In his 1976 book 100 Numerical Games, French puzzle maven Pierre Berloquin asks whether it’s possible to construct a magic square using the first nine prime numbers (here counting 1 as prime):

1 2 3 5 7 11 13 17 19

Is it?

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No. The numbers total 78, so in a successful magic square each row and column would total a third of that, or 26. But only one of the primes is even, which means that in two rows and two columns we’d have to produce an even number by adding three odd numbers, which is impossible.

June 14, 2017June 18, 2017 | Puzzles

Black and White

matthäus chess problem

By Hans Georg Matthäus. White to mate in two moves.

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If the rook weren’t on f2 then White could immediately give mate with the knight. Unfortunately, moving the rook out of the way gives Black time to play 1. … Bd4, and now White’s plan doesn’t work — if the knight checks the king then the bishop can simply capture it. The answer is 1. Rf6! This stops the bishop from reaching d4, and Black has no way to stop 2. Nf2#.

June 6, 2017June 18, 2017 | Puzzles