A PuzzleCritic Original:
The Museum of Mathematical Mysteries houses a peculiar collection of coins, each shaped like a polygon, with the two largest proper factors of n inscribed on either face, where n is the number of edges the given coin possesses. For example, there is a hexagonal coin on which the numbers 2 and 3 are inscribed. The museum’s curator examines one of the coins and sees the number 15 inscribed on one face. Determine all numbers that might appear on the other face.
Here I present one of my favourites; a problem of rare invention. From the 1998 Tournament of the Towns:
(Two-person case) Two robbers stole a bag of coins from a merchant. Each coin is worth an integer number of pennies. It is known that if any single coin is removed from the bag, then the remaining coins can be divided fairly among the two robbers (that is, they both get coins with the same total value in pennies).
Prove that after one coin is removed, the number of coins remaining is even.
(General case) A gang of robbers stole a bag of coins from a merchant. Each coin is worth an integer number of pennies. It is known that if any single coin is removed from the bag, then the remaining coins can be divided fairly among the robbers (that is, they all get coins with the same total value in pennies).
Prove that after one coin is removed, the number of coins remaining is divisible by the number of robbers.
It’s here! My new book of puzzles Elastic Numbers is now available on Amazon.
I hope you enjoy solving the problems as much as I enjoyed creating them.
My new book of maths puzzles is on its way! It’s packed full of interesting problems to sink your teeth into. I’ll post an update as the launch approaches.
In the meantime, here is a fantastic problem from the USA:
It is given that is a 604-digit number beginning with a 1. How many of the numbers begin with a 4?
A PuzzleCritic Original:
There are 999 stones in a pile. Amisi and Boaz take it in turns removing either 3 or 5 stones from the pile, with Amisi going first, until no more moves are possible. The last player to make a move wins. Which player can guarantee victory?
From the 2004 Tournament of Towns:
Each day, the price of the shares of the corporation “Soap Bubble, Limited” either increases or decreases by n%, where n is an integer such that 0<n<100. The price is calculated with unlimited precision. Does there exist an n for which the price can take the same value twice?
Having spent nine days in Hamburg at a maths conference, I return to present what is to date my favourite ever maths problem. From the Tournament of the Towns:
Let n be a positive integer. Consider the largest odd factor of each of the numbers n+1, n+2, …, 2n. Prove that their sum is n2.