Category Archives: Puzzles and Problems

Parallelogram Problem

Catriona Shearer has come up with another challenging but elegant geometric problem. In some ways, it is similar to the famous Russian Coffin Problems that have an obvious solution—once you see it—but initially seem impenetrable. I really marvel at Catriona Shearer’s ability to come up with these problems.

“What’s the area of the parallelogram?”

See the Parallelogram Problem

Counterfeit Coin in Base 3

Futility Closet presented a nifty method of solving the “counterfeit coin in 12 coins” problem in a way I had not seen before by mapping the problem into numbers in base 3. It wasn’t immediately clear to me how their solution worked, so I decided to write up my own explanation.

Futility Closet: “You have 12 coins that appear identical. Eleven have the same weight, but one is either heavier or lighter than the others. How can you identify it, and determine whether it’s heavy or light, in just three weighings in a balance scale? This is a classic puzzle, but in 1992 Washington State University mathematician Calvin T. Long found a solution ‘that appears little short of magic.’ ”

See Counterfeit Coin in Base 3.

Missing Pages Puzzle

Setting aside my chagrin that the following problem was given to pre-university students, I initially found the problem to be among the daunting ones that offer little information for a solution. It also was a bit “inelegant” to my way of thinking, since it involved considering some separate cases. Still, the end result turned out to be unique and satisfying (Talwalkar’s Note 2 was essential for a unique solution, since the problem as stated was ambiguous).

“Kshitij from India sent me this problem from the 1994 India Regional Mathematics Olympiad.
A leaf is torn from a paperback novel. The sum of the numbers on the remaining pages is 15000. What are the page numbers on the torn leaf?
Note 1: a ‘leaf’ means a single sheet of paper.
Note 2: the quoted problem is actual wording from the competition. But let me add an important detail: the book is numbered in the usual sequential way starting with the first page as page 1.” See the Missing Pages Puzzle.

River Crossing

This is a riff on a classic problem, given in Challenging Problems in Algebra.

“N. Bank and S. Bank are, respectively, the north and south banks of a river with a uniform width of one mile. Town A is 3 miles north of N. Bank, town B is 5 miles south of S. Bank and 15 miles east of A. If crossing at the river banks is only at right angles to the banks, find the length of the shortest path from A to B.

Challenge. If the rate of land travel is uniformly 8 mph, and the rowing rate on the river is 1 2/3 mph (in still water) with a west to east current of 1 1/3 mph, find the shortest time it takes to go from A to B. [The path across the river must still be perpendicular to the banks.]” See the River Crossing.

Chalkdust Triangle Problem

The issue 7 of the Chalkdust mathematics magazine had an interesting geometric problem presented by Matthew Scroggs.

“In the diagram, ABDC is a square. Angles ACE and BDE are both 75°. Is triangle ABE equilateral? Why/why not?”

I had a solution, but alas, the Scroggs’s solution was far more elegant. See the Chalkdust Triangle Problem.

Star Sum of Angles

This problem posted by Presh Talwalkar offers a variety of solutions, but I didn’t quite see my favorite approach for such problems. So I thought I would add it to the mix.

“Thanks to Nikhil Patro from India for suggesting this! What is the sum of the corner angles in a regular 5-sided star? What is a + b + c + d + e = ? Here’s a bonus problem: if the star is not regular, what is a + b + c + d + e = ?”

See Star Sum of Angles

More Pool

This is another UKMT Senior Challenge problem, this time from 2006.

“A toy pool table is 6 feet long and 3 feet wide. It has pockets at each of the four corners P, Q, R, and S. When a ball hits a side of the table, it bounces off the side at the same angle as it hit that side. A ball, initially 1 foot to the left of pocket P, is hit from the side SP towards the side PQ as shown. How many feet from P does the ball hit side PQ if it lands in pocket S after two bounces?”

Pool Partiers should have no difficulty solving this. See More Pool.