Prime Chain — draw paths whose neighbours share no prime factor

What is Prime Chain — draw paths whose neighbours share no prime factor

Prime Chain gives you a six-by-six grid holding every number from 2 to 30 once, plus seven repeats, and a single rule for joining them. Press a cell and drag through cells that touch it edge to edge; each new number must be coprime with the one immediately before it, sharing no prime factor at all. Under every number sits a row of coloured dots, one for each distinct prime that divides it: red for 2, blue for 3, yellow for 5. Two cells link only when their dots have no colour in common, and legal continuations glow with a green border.

Releasing the path, or pressing End Chain, banks it. A chain pays its sum multiplied by its length squared, so eight cells adding to 80 are worth 5120 points, while the same eight split into two chains of four pay 1280. Banked cells fade out and leave the round, which ends the moment no two surviving neighbours are coprime, usually with a scatter of stranded even numbers that can never touch. Three hints per round point at a legal move, Undo steps back one cell, and a one-cell chain costs nothing to abandon. The four readouts above the grid track the round score, your best, the live chain length and your day streak.

How to play

1. Press any cell to open a chain, then drag through cells that share an edge with it; a green border marks every legal next step.
2. The dots under a number are its distinct prime factors, and two cells connect only when they share no dot colour at all.
3. Press End Chain to bank the path: it pays the sum of its numbers times the length squared, and those cells leave the board.
4. Open the next chain elsewhere. The round ends when no two adjacent surviving cells are coprime, and the round score goes to the ranking.

Strategies that work

• Two even numbers can never follow each other, since 2 divides both, and about half the grid is even. Every red dot walls off another red dot, so a long path has to weave odd cells between the evens. Read the reds first and treat the odd numbers as your bridges.
• Only consecutive cells are tested, never the whole chain. So 12, then 25, then 18 is legal even though 12 and 18 both carry 2 and 3, because 25 clashes with neither of them. A large odd number parked between two multiples of six collects both in one path.
• Length is squared and the sum is not, so extra cells beat large cells. Four cells adding to 40 pay 640; eight cells adding to 80 pay 5120, eight times as much for twice the material. Never bank a chain that still has a green neighbour waiting.
• 17, 19, 23 and 29 have no other multiple on the board, since twice 17 is already 34, so each of them joins to every other value and rescues a boxed-in path. Their opposite is 30, the only cell with three dots: it accepts seven partners, 7, 11, 13, 17, 19, 23 and 29.
• Count dots, not size. 16 carries one red dot and pairs with every odd value from 3 to 29, fourteen of them; 24 carries red and blue and pairs with nine. The round dies when all the survivors share factors, so clear the crowded 6, 12, 18 and 24 early.

A little history

Prime Chain was written for this site, and its rule comes straight from elementary number theory. Every whole number above 1 breaks into primes in exactly one way, a result approached by Euclid in the Elements and stated in full by Gauss in the Disquisitiones Arithmeticae of 1801. Two numbers are coprime when those factorisations share nothing, which is what Euclid's algorithm for the greatest common divisor measures. Primes also thin out as numbers grow — there are 25 below 100 — so on a board stopping at 30 the small ones do nearly all of the blocking.

Frequently asked questions