A PUZZLE MADE OF EVEN AND ODD · FROM THE KIT
Every lantern carries a number and burns a flame for each prime it holds an odd number of times. Light lanterns until every flame is out: then the numbers you lit multiply to a perfect square.
Light a lantern to begin.
A number is a perfect square when every prime in it appears an even number of times. Write a 1 for odd and a 0 for even and each lantern is a row of bits; lighting two lanterns adds their rows, and in this arithmetic adding is XOR. A set of rows that adds to all zeros is a set of numbers whose product is a square. The line under the lanterns is the square split, product = r² + rem, and the remainder is zero exactly when you have won.
The row is dealt by indexed randomness (the same row for the same name on every machine), the split is judged by the ruler land's square remainder, and the hint is the null space of the lanterns' rows, found by the same GF(2) elimination as Lars Logic.
The Square Lanterns · by Lars Rocha · rebuilt from sitoolz: indexrng, rulerland and logic.