## The Grid

A PuzzleCritic Original:

An empty 6-by-6 grid of squares is filled with numbers as follows. The first row contains the numbers {1,2,3,4,5,6} in some order. In each subsequent row, the kth number is equal to the position of the number k in the row above. For how many such grids are the first and last rows identical?

## The Integer Shuffle

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?

## The Coin Collection

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.

## A Game of Stones

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?

## A Strange Die

A PuzzleCritic Original:

A cube has a different positive integer painted on every face, so that every pair of numbers on adjacent faces has a sum divisible by 7. Let T be the sum of all the numbers that appear. Find, with proof, the smallest possible value of T.

## The answer in the problem

A PuzzleCritic Original:

Let x be the answer to this problem, where x is a positive integer, and let y be the sum of its digits. Calculate 2x-2y.

## PuzzleCritic Originals: A strange festival

Here is a problem of my own invention; I hope you enjoy it:

At a festival, every guest presented a gift to every older guest. The ten youngest guests combined gave thirteen times as many gifts as they received. What was the total number of gifts presented at the festival?