Rectangles and Primes

Applets have been created to help the viewer arrange any number of objects into rectangular patterns.

A game has been created asking the user to pack a given number of chocolates into rectangular boxes having at least two rows or two columns, filling each box completely. (The player gains no points by choosing the trivial solutions of boxes with exactly one row or one column.) The user clicks on the number of columns in the box and the computer tries to fill the box completely. If the number of columns is not a divisor of the given number of chocolates, it is impossible to fill the box completely; the computer shows the partially filled box with the leftover pieces. Otherwise there may be several ways to fill the box, and each filled box changes color. Positive numerical scores are given for success in filling a box completely and nontrivially, negative scores for failure.

The game can be played at different levels. At level k the number of chocolates is less than the square of k. A tutorial will teach that at level k one should try clicking only on the smallest primes less than k. If this succeeds, then one can find another solution (the transpose). This illustrates the well-known principle that in checking whether a number less than N is prime or composite, it suffices to check divisibility by the smallest primes less than the square root of N.

If one plays very badly and highly negative scores accumulate, the tutorial will make suggestions for improvement.