Category Archives: Puzzles and Problems

Three Dutchmen Puzzle

Presh Talwalkar presented an interesting puzzle that originated in the Ladies’ Diary of 1739-40, was recast by Henry Dudeney in 1917, and further modified using American money.

“Each of three Dutchmen, named Hendrick, Elas, and Cornelius has a wife. The three wives have names Gurtrün, Katrün, and Anna (but not necessarily matching the husband’s names in that order). All six go to the market to buy hogs.

Each person buys as many hogs as he or she pays dollars for one. (1 hog costs $1, 2 hogs are $2 each, 3 hogs cost $3 each, etc.) In the end, each husband has spent $63 more than his wife. Hendrick buys 23 more hogs than Katrün, and Elas 11 more than Gurtrün. Now, what is the name of each man’s wife?”

See the Three Dutchmen Puzzle for solutions.

The Umbrella Problem

This is a rather mind-boggling problem from the 1947 Eureka magazine.

“Six men, A, B, C, D, E, F, of negligible honesty, met on a perfectly rough day, each carrying a light inextensible umbrella. Each man brought his own umbrella, and took away—let us say “borrowed”—another’s. The umbrella borrowed by A belonged to the borrower of B’s umbrella. The owner of the umbrella borrowed by C borrowed the umbrella belonging to the borrower of D’s umbrella. If the borrower of E’s umbrella was not the owner of that borrowed by F, who borrowed A’s umbrella?”

Answer.

See the Umbrella Problem for solutions.

Horses to Qi

This is a challenging problem from the c.100AD Chinese mathematical work, Jiǔ zhāng suàn shù (The Nine Chapters on the Mathematical Art) found at the MAA Convergence website.

“Now a good horse and an inferior horse set out from Chang’an to Qi.  Qi is 3000 li from Chang’an.  The good horse travels 193 li on the first day and daily increases by 13 li; the inferior horse travels 97 li on the first day and daily decreases by ½ li.  The good horse reaches Qi first and turns back to meet the inferior horse.  Tell: how many days until they meet and how far has each traveled?”

The solution involves common fractions, which the Chinese were already adept at using by 100 BC.

Answer.

See Horses to Qi for a solution.

Making Arrows

This is an interesting problem from 180 BC China.

“In one day, a person can make 30 arrows or fletch [put the feathers on] 20 arrows.  How many arrows can this person both make and fletch in a day?”

It turns out the solution to this problem led me into the history of numerator/denominator (aka common) fractions, a subject I had been finding difficult to track down.

Answer.

See Making Arrows for a solution.

Hjelmslev’s Theorem

I came across this remarkable result in Futility Closet:

“On each of these two black lines is a trio of red points marked by the same distances.  The midpoints of segments drawn between corresponding points are collinear.

(Discovered by Danish mathematician Johannes Hjelmslev.)”

This result seems amazing and mysterious.  I wondered if I could think of a proof.  I found a simple approach that did not use plane geometry.  And suddenly, like a magic trick exposed, the result seemed obvious.

See Hjelmslev’s Theorem

Classic Geometry Paradox

Coming across this classic geometric paradox recently in Futility Closet motivated me to write down its solution in detail.

“Where did the empty square come from?”

In any case, this is the canonical example for why I avoid visual geometric proofs—you can be so easily fooled.  Real proofs require plane or analytic geometry arguments.

See the Classic Geometry Paradox

(Update 9/14/2024) Penn & Teller – Fool Us – Magic Trick
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