One of a Kind

You’re about to roll five regular dice. Which is more likely, rolling no sixes or rolling exactly one six?

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The chance of rolling no sixes is 5/6 × 5/6 × 5/6 × 5/6 × 5/6 The chance of rolling exactly one six is 1/6 × 5/6 × 5/6 × 5/6 × 5/6 … times 5, because the six can appear on any one of the five dice. So the probabilities are equal.

May 17, 2012February 4, 2013 | Puzzles

Sweet and Low

At age 10, Nob Yoshigahara won the grand prize in a newspaper competition with this puzzle:

Given a sack of sugar, an unbalanced scale, and two 5-pound weights, measure exactly 10 pounds of sugar.

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Put both weights in one pan and pour sugar into the other until it balances. Then remove the weights and pour sugar into the first pan until it balances again. The first pan will contain exactly 10 pounds of sugar.

May 16, 2012February 4, 2013 | Puzzles

Leftovers

A problem from Litton Problematical Recreations, which attributes it to Fermat circa 1635:

What is the remainder upon dividing 5^999,999 by 7?

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When successive powers of 5 are divided by 7, the remainders form a repeating series: 5¹ / 7 = 0 remainder 5 5² / 7 = 3 remainder 4 5³ / 7 = 17 remainder 6 5⁴ / 7 = 89 remainder 2 5⁵ / 7 = 446 remainder 3 5⁶ / 7 = 2232 remainder 1 5⁷ / 7 = 11160 remainder 5 5⁸ / 7 = 55803 remainder 4 5⁹ / 7 = 279017 remainder 6 5¹⁰ / 7 = 1395089 remainder 2 … The 999,999th term of the series is 6.

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May 14, 2012February 4, 2013 | Puzzles

Red White and Blue

I place four balls in a hat: a blue one, a white one, and two red ones. Now I draw two balls, look at them, and announce that at least one of them is red. What is the chance that the other is red?

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It’s 1 in 5. Six possible pairings can be drawn from the hat: red 1 / red 2 red 1 / white red 2 / white red 1 / blue red 2 / blue white / blue We know that the last has not been drawn. That leaves 5 possible combinations, so the possibility of red 1 / red 2 is 1 in 5. “Many people cannot accept that the solution is not 1 in 3,” writes Erwin Brecher, “and of course it would be, if the balls had been drawn out separately and the color of the first ball announced as red before the second had been drawn out. However, as both balls had been drawn together, and then the color of one of the balls announced, then the above solution, 1 in 5, must be the correct one.”

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May 13, 2012February 4, 2013 | Puzzles

Black and White

dehler chess problem

By O. Dehler, Tidskrift för Schack, 1928. White to mate in two moves.

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1. Qh7 (K moves) 2. e8=Q#.

May 11, 2012February 4, 2013 | Puzzles

Sand Reckoning

Using only a 4-minute hourglass and a 7-minute hourglass, how can you measure 9 minutes?

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Turn over both hourglasses. When the 4-minute glass has run out, turn it over. When the 7-minute glass finishes, turn it over and let it run until the 4-minute glass has run out. At this point we’re at the 8-minute mark and the lower bulb of the 7-minute glass contains 1 minute’s worth of sand. Turn it over to measure the final minute.

May 7, 2012February 4, 2013 | Puzzles

Spin Control

We’re playing Russian roulette. The revolver has six chambers, all empty. I put bullets in two adjacent chambers, spin the cylinder, hold the gun to my head, and pull the trigger. It clicks. Now it’s your turn. Before pulling the trigger, you can choose to spin the cylinder again or leave it as it is. Which is better?

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It’s better not to spin it. Two of the six chambers are loaded, so my chance of surviving was 4/6. But you know that the hammer fell on an empty chamber. Of the four empty chambers, only one is followed by a loaded one. So if you simply pull the trigger, you have a 3/4 chance of surviving. That’s better than 4/6, so you’re better off not spinning the cylinder.

May 6, 2012February 4, 2013 | Puzzles

Secret Entrance

Puzzle maven Noboyuki Yoshigahara noticed this sign on the glass door of an Osaka restaurant:

secret entrance puzzle

What is its meaning?

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The words PULL and PUSH were painted on opposite sites of the door.

May 4, 2012February 4, 2013 | Puzzles

Black and White

t.r. dawson chess problem

By T.R. Dawson, Chess Amateur, 1920. White to mate in two moves.

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1. b8=N! Kxe8 2. Qg8#

May 3, 2012March 1, 2013 | Puzzles

Making Way

A man is 4/7 across a train trestle when he sees a train coming. To get off the trestle, he can run toward the train or away from it. As it happens, in either case he’ll reach safety just as the locomotive passes him. If he runs at 20 kph, how fast is the train going?

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If he runs toward the train, he’ll cover the remaining 3/7 of the trestle and meet the train at the far end. This means that if he runs away from the train, he’ll have covered 3/7 of the trestle (with 1/7 to go) when the train reaches the far end. He’ll then run the remaining 1/7 while the locomotive crosses all 7/7 and passes him. So the train is traveling at 7 times his speed, or 140 kph.

May 2, 2012February 1, 2013 | Puzzles