Category Archives: challenges

A Fibonacci pattern

Posted on May 13, 2011 by Brent

Recall the Fibonacci numbers, , the sequence of numbers beginning with where each subsequent number is the sum of the previous two: Try this: pick any Fibonacci number. Square it. Now, look at the two Fibonacci numbers on either side … Continue reading →

Posted in algebra, arithmetic, challenges, fibonacci, pattern, sequences | Tagged , , | 4 Comments

An area paradox

Posted on May 2, 2011 by Brent

Here’s a fun paradox which has been around for quite a while and was apparently a favorite of Lewis Carroll. As you can verify for yourself, the two figures above are composed of two different rearrangements of the same four … Continue reading →

Posted in challenges, geometry, paradox, pictures | Tagged , , | 18 Comments

Triangular number equations via pictures

Posted on March 24, 2011 by Brent

The other day I was fiddling around a bit with triangular numbers. By only drawing pictures I was able to come up with the following triangular number equations, where denotes the th triangular number (that is, the number of dots … Continue reading →

Posted in challenges, pictures, proof | Tagged , , , | 8 Comments

More cookies

Posted on July 27, 2010 by Brent

I recently received the following interesting problem from Shadowcat, which is a generalization of the cookie problem I’ve written about previously. We again want to count the ways to distribute identical cookies to non-identical students, with the twist that we … Continue reading →

Posted in arithmetic, challenges, counting | Tagged , , | 9 Comments

Optimal change-carrying

Posted on June 24, 2010 by Brent

Recently Michael left the following challenge in a comment: I’ve been trying to optimize my change-carrying habits. What is the smallest amount of quarters, dimes, nickels and pennies one can carry while still being able to give perfect change (two … Continue reading →

Posted in arithmetic, challenges, counting | Tagged , | 11 Comments

The broken weight problem: solutions and further exploration

Posted on May 11, 2010 by Brent

First of all, let me say to all my readers how fantastic it felt to post a puzzle, after not posting anything for two months, and get eighteen thoughtful, insightful comments in just three days; it’s every blogger’s dream. You … Continue reading →

Posted in arithmetic, challenges, number theory, solutions | Tagged , , , , | 6 Comments

The broken weight problem

Posted on May 1, 2010 by Brent

Here’s a fantastic problem I recently heard. Apparently it was first posed by Claude Gaspard Bachet de Méziriac in a book of arithmetic problems published in 1612, and can also be found in Heinrich Dorrie’s 100 Great Problems of Elementary … Continue reading →

Posted in arithmetic, challenges, number theory, puzzles | Tagged , | 22 Comments

The haybaler

Posted on December 16, 2009 by Brent

At Penn Alexander’s math club yesterday, the students worked on a fun puzzle that I’d never seen before. It goes like this: You have five bales of hay. For some reason, instead of being weighed individually, they were weighed in … Continue reading →

Posted in arithmetic, challenges, logic, puzzles, teaching | Tagged , , | 11 Comments

Who Am I?

Posted on November 29, 2009 by Brent

An excellent puzzle from JD2718: There are five true and five false statements about the secret number. Each pair of statements contains one true and one false statement. Find the trues, find the falses, and find the number. 1a. I … Continue reading →

Posted in challenges, logic, number theory, puzzles, teaching | Tagged , , | 13 Comments

Hyperbinary conjecture seeking proof for a good time, long walks on the beach

Posted on October 12, 2009 by Brent

Here’s the latest progress on the hyperbinary sequence. We’re trying to figure out the inverse relation of the function : given a particular number , where does it occur in the hyperbinary sequence? That is, what are the values of … Continue reading →

Posted in challenges, pattern, people, proof, sequences | Tagged , , , | 4 Comments