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

Black and White

loquin chess problem

By François Paul Loquin. White to mate in two moves.

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1. Kf1! Kxh3 2. Bf5# From La Régence, 1849.

May 15, 2017May 28, 2017 | Puzzles

Podcast Episode 152: Lateral Thinking Puzzles

Here are five new lateral thinking puzzles to test your wits and stump your friends — play along with us as we try to untangle some perplexing situations using yes-or-no questions.

See full show notes …

May 1, 2017September 30, 2017 | Podcast · Puzzles

Black and White

gottschall chess problem

By Hermann Von Gottschall. White to mate in two moves.

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1. Bh1! Kxh1 2. Rc1#

April 24, 2017May 7, 2017 | Puzzles

In a Word

ullage
n. the amount a container lacks of being full

Given a 5-gallon jug, a 3-gallon jug, and a limitless supply of water, how can you measure out exactly 4 gallons?

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Fill the 5-gallon jug and use it to fill the 3-gallon jug. That leaves 2 gallons in the 5-gallon jug. Empty the 3-gallon jug and transfer the 2 gallons from the 5-gallon jug to the 3-gallon jug. Fill the 5-gallon jug and use it to top off the 3-gallon jug. That requires exactly 1 gallon, leaving 4 gallons in the 5-gallon jug. You can also solve this geometrically, or you can let John McClane solve it for you: 04/23/2017 UPDATE: Two readers give a more efficient solution: Fill the 3-gallon jug and empty it into the 5-gallon jug. Fill the 3-gallon jug again and use it to top off the 5-gallon jug. This leaves 1 gallon in the 3-gallon jug. Empty the 5-gallon jug. Transfer the remaining gallon from the 3-gallon jug to the 5-gallon jug. Fill the 3-gallon jug again and pour it into the 5-gallon jug. This solution requires discarding 5 gallons altogether, where mine required discarding 6. (I think this is probably the alternate solution that Grossman’s method will obtain, but I haven’t had time to check.) (Thanks, Chris and Marty.)

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April 20, 2017April 30, 2017 | Language · Puzzles

The Cherries Puzzle

ozanam cherries puzzle

A classic puzzle from Jacques Ozanam’s Recreations Mathematiques et Physiques, 1723. Two slits (CD) and two holes (EF) are cut in a slip of paper, and a cherry stem is suspended as shown. The cherries are too large to fit through the holes. How can you free the stem and its cherries intact from the slip?

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Fold the slip and feed the strip through both holes as shown. Now you can draw one cherry through loop H.

April 14, 2017April 23, 2017 | Puzzles