A stunner from the Baltic Way:
A family wears clothes of three colours: red, blue and green, with a separate identical laundry bin for each colour. At the beginning of the first week, all the bins are empty. Each week, the family generates 10kg of laundry (the proportion of each colour might vary week to week). The laundry is sorted by colour and placed in the bins. Next, the heaviest bin (only one if there are several) is emptied and its contents washed.
What is the minimum requirement (in kg) of each bin?
From the Netherlands Junior Maths Olympiad:
There is a kangaroo in the centre of each square of a 5-by-5 grid of squares. Lightning strikes, and each kangaroo simultaneously jumps one square up, down, left or right. What is the greatest possible number of empty squares that could remain?
A PuzzleCritic Original:
A sequence contains every positive integer exactly once, and no other terms. Must there exist, somewhere in the sequence:
(i) an odd number immediately followed by an even number;
(ii) a multiple of two immediately followed by a multiple of three?
This puzzle, which I posted to twitter (@puzzlecritic) a few weeks ago, is one of my variations on a problem that appeared in a mathematical discussion group on LinkedIn:
Find three different positive integers such that the sum of any two is a square.
From the 2003 Tournament of the Towns:
Each term in a sequence of positive integers is obtained from the previous term by adding it to its largest digit. What is the greatest possible number of successive odd terms in such a sequence?
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.