Category Archives: challenges

A Fibonacci pattern

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

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

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

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

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

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

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