Category Archives: Puzzles and Problems

Another Blank Clock

This is a blank clock puzzle by Heinrich Hemme from Scientific American math puzzles.

“The hour, minute and second hands of this clock are all the same length and move smoothly in a circle. The dial contains hour and minute markers, but the numbers are missing. Therefore, it’s impossible to tell which one of the 12 hour markers belongs to the 12. The two hands on the left are positioned exactly on hour markers, and the hand on the right is positioned between a minute and an hour marker. What time does the clock show?”

Answer.

See Another Blank Clock for solutions.

Zigzag Line Problem

This is an interesting Talwalkar problem from 2021.

“Thanks to Lucas from Brazil for the suggestion! This problem is adapted from a Kangaroo Maths competition in 2020 for grades 9-10 (around 15 years old).

A circle has diameter AB. A line starts at A and zig-zags exactly 4 times between the diameter and the circumference until it ends at B, as shown.

If each of the angles the line makes with the diameter has the same measure α, what is α equal to?”

Answer.

See the Zigzag Line Problem for a solution.

Knock the Bottle Problem

This is a problem from Barry Leung’s Math Games.

“Larry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge.  Larry throws first. The winner is the first person to knock the bottle off the ledge.  At each turn the probability that a player knocks the bottle off the ledge is ½, independently of what has happened before.  What is the probability that Larry wins the game?”

Answer.

See Knock the Bottle Problem for a solution.

Clockwise Ant Puzzle

This is actually a travel problem masquerading as a clock puzzle from Futility Closet.

“A problem by Argentinian puzzlist Jaime Poniachik, from the February 1992 issue of Games magazine:

An ant crawls onto a clock face at the 6 mark just as the minute hand is passing 12. She begins crawling counterclockwise around the face’s circumference at a uniform speed. When the minute hand passes her, she reverses course and crawls clockwise without changing her speed. Forty-five minutes after her first encounter with the minute hand, it passes her a second time and she departs. How much time did she spend on the clock face?”

Answer.

See Clockwise Ant Puzzle for solutions.

A Pair of Pretty Perimeter Puzzles

Here are two puzzles from Alex Bellos’s Monday puzzles.

“Ring it.  Each region has a perimeter given by its enclosed number. What is the length just along the edge of the entire figure?

Round the block.  Assuming all corners are right angles, what is the perimeter?

Today’s puzzles all come from … the Hyde Park Math Zine!  This delightful publication is written in pen on a single folded sheet of paper, has a print run of 30 copies, and is distributed in the neighbourhood of Hyde Park in Austin, Texas.

Fanzine culture is well established in sports and music. Math educator Kevin Gately thought the format would work for puzzles too. “It dawned on me that there might be people in my community who find the novelty of a hyper-local math zine to be amusing and/or curious,” he said. And it seems there are.

Each issue of HPMZ presents three problems, with easily understandable answers, and let’s not forget the cover artwork!  Gately’s puzzles are mostly taken from other sources, and tweaked. Here are [two] that took my fancy.”

Answer 1 ________  Answer 2

See A Pair of Pretty Perimeter Puzzles for solutions.

Language Students Problem

This is a problem from the 2001 American Invitational Mathematics Exam (AIME).

“Each of the 2000 students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between 80 percent and 85 percent of the school population, and the number who study French is between 30 percent and 40 percent. Let m be the smallest number of students who could study both languages, and let M be the largest number of students who could study both languages. Find M – m.”

Answer.

See Language Students Problem for a solution.

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