Black and White

By J.H. Finlinson. White to mate in two moves.

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1. Qg6 and White will mate with 2. Qb1, 2. Rb1, 2. Na7, or 2. Rxc5. From J. Paul Taylor, Chess Chips, 1878.

October 5, 2012February 7, 2013 | Puzzles

Injured List

Another puzzle from Henry Dudeney:

“It is a glorious game!” an enthusiast was heard to exclaim. “At the close of last season, of the footballers of my acquaintance, four had broken their left arm, five had broken their right arm, two had the right arm sound, and three had sound left arms.” Can you discover from that statement what is the smallest number of players that the speaker could be acquainted with?

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The smallest possible number of men is seven. They could be accounted for in three different ways. (1) Two with both arms sound, one with broken right arm, and four with both arms broken. (2) One with both arms sound, one with broken left arm, two with broken right arm, and three with both arms broken. (3) Two with left arm broken, three with right arm broken, and two with both arms broken. But if every man was injured, the last case is the only one that would apply.

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October 2, 2012February 7, 2013 | Puzzles

The Dwarfs Problem

From the 1977 all-Soviet-Union Mathematical Olympiad:

Seven dwarfs are sitting at a round table. Each has a cup, and some cups contain milk. Each dwarf in turn pours all his milk into the other six cups, dividing it equally among them. After the seventh dwarf has done this, they find that each cup again contains its initial quantity of milk. How much milk does each cup contain, if there were 42 ounces of milk altogether?

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Since the final state matches the initial state, we can imagine this process going on continuously. Consider the dwarf whose cup contains the smallest amount of milk just before he begins pouring. Call him D, and call this quantity of milk a ounces. Since D’s cup contains the least milk before pouring, then each other dwarf has an equal or greater amount in his own cup before pouring, and thus gives D at least a/6 ounces. This would leave D with at least a ounces in his own cup just before pouring; the only way he can receive precisely a ounces, as we know he does, is if each other dwarf gives him precisely a/6 ounces and no more, i.e., that each cup, just before pouring, contains the same quantity of milk. So after any given pouring the cups contain a, 5/6a, 4/6a, 3/6a, 2/6a, 1/6a, and 0 ounces; if there are 42 ounces in total then this works out to 12, 10, 8, 6, 4, 2, and 0 ounces.

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October 1, 2012February 7, 2013 | Puzzles

Tiling Task

tiling task 1

We’ve removed two squares from this 7×8 grid, so that it numbers 54 squares. Can it be covered orthogonally with tiles like the one at right, each of which covers exactly three squares?

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No. Number the squares as shown. Now each tile, however it’s placed, must cover a 1, a 2, and a 3. But the grid contains 19 1s, 18 2s, and 17 3s. So the task is impossible. From Pierre Berloquin, The Garden of the Sphinx, 1981.

September 30, 2012February 7, 2013 | Puzzles

The Two Poles

dudeney poles puzzle

A puzzle by Henry Dudeney:

A man planted two poles upright in level ground. One pole was six and a half feet and the other seven feet seven inches above ground. From the top of each pole he tied a string to the bottom of the other — just where it entered the ground. Now, what height above the ground was the point where the two strings crossed one another? The hasty reader will perhaps say, “You have forgotten to tell us how far the poles were apart.” But that point is of no consequence whatever, as it does not affect the answer!

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One pole was 78 inches high, and the other 91 inches. Multiply these two numbers together, and also add them together. Then divide the first result by the second, and you get 42 inches as the height above ground of the point where the two strings intersect, no matter how far the poles may be apart.

September 28, 2012February 7, 2013 | Puzzles

Black and White

By R.F. Bennett. White to mate in two moves.

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1. Bc7 and the queen will mate. From J. Paul Taylor, Chess Chips, 1878.

September 27, 2012February 7, 2013 | Puzzles

Out of Rank

beasley chess puzzle

A puzzle from John Beasley’s The Mathematics of Games (2006): Black has just moved. What is the smallest number of moves that can have been played in this game?

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Black can have moved only his knights and his rooks. The knights began on squares of different colors, and a moving knight lands on light and dark squares alternately, so between them the two knights have made an even number of moves (even if they’ve exchanged places). While the knights were abroad the rooks may have shifted back and forth a bit, but they’re on their home squares now and so have made an even number of moves themselves (possibly zero). So Black has made an even number of moves altogether. If Black has made an even number of moves, then so has White. But this presents a problem: White’s knights and rooks have moved an even number of times, but the f-pawn’s position on f3 means that White has made an additional move, giving an odd total. In order to make the total even we must find an odd quantity of additional moves that White can have made, and they can’t have been made with pawns, knights, or rooks. The simplest solution is that the king has made seven moves, e.g., e1-f2-e3-f4-g4-g3-f2-e1. So the answer is that each player has made at least eight moves: The shortest game that could produce the position above with White to play begins with 1. f3 and then the white king makes seven moves while Black marks time with his knights (and perhaps his rooks).

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September 25, 2012February 7, 2013 | Puzzles

Drinking Problem

I have a 16-ounce bottle of wine and want to make it last as long as possible, so I establish the following plan: On the first day I’ll drink 1 ounce of wine and refill the bottle with water. On the second day I’ll drink 2 ounces of the mixture and refill the bottle with water. On the third day I’ll drink three ounces of the mixture and again refill the bottle with water. If I continue until the bottle is empty, how many ounces of water will I have drunk?

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It’s easy to get bogged down in calculating proportions, but there’s a simpler solution. If I continue until the bottle is empty, then eventually I will have drunk all the water that I’ve added to the bottle. I add 1 ounce on the first day, 2 on the second, and so on to the 15th day. (I drink off the entire bottle on the 16th day, so no water is added then.) So in all I’ll have drunk 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 = 120 ounces of water. (Another solution is to calculate that in the end I’ve drunk 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 = 136 ounces in total, and then deduct the 16 ounces of wine.)

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September 23, 2012February 6, 2013 | Puzzles

Weighty Matters

I’ve accidentally turned the calibration dial on my bathroom scale, so its readings are skewed by a consistent amount. Apart from that it works fine, though. When I stand on the scale it reads 170 pounds, when my wife stands on it it reads 130, and when we stand on it together it reads 292 pounds. How should I adjust the scale?

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When I’m standing on the scale and my wife joins me, the difference in the readings will show us her true weight, 292 – 170 = 122 pounds. So the scale is set 130 – 122 = 8 pounds too high.

September 22, 2012February 6, 2013 | Puzzles

Black and White

From Alexander H. Robbins, A Collection of Chess Problems, 1887. White to mate in two moves.

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The quiet 1. Qb3 threatens six different mates.

September 21, 2012February 6, 2013 | Puzzles