Diophantine Workshop Game
Explore integer solutions to equations of the form ax + by = n.
Choose Practice or Daily to begin.
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Instructions, device progress and accessibility
Choose Easy, Normal or Hard and select Practice or a daily puzzle based on the UTC date. Enter answers, then select Check answer. Hints and Undo mark a round as assisted. Best scores and progress are stored only in this browser; no account is needed.
Local progress is checked when play begins.
How to play Diophantine Workshop
Find every nonnegative whole-number pair (x, y) that satisfies the displayed equation. Enter how many pairs work, and the pair with the largest possible x. Use systematic enumeration rather than guessing.
Worked example
For 2x + 3y = 12, the nonnegative solutions are (0,4), (3,2), and (6,0). There are 3 solutions and the largest x is 6.
Math formulas and key skills
For ax + by = n with positive integer a and b, test x = 0,1,…,floor(n/a). A valid y = (n − ax)/b must be a nonnegative integer.
Avoid common mistakes
Do not count negative or fractional values of x or y. Count ordered pairs only once, and check both coefficients.
Frequently asked questions
What is a Diophantine equation?
An equation whose solutions are restricted to integers; these puzzles further require nonnegative integers.
How can I be sure I found every pair?
Enumerate x from zero to its maximum possible value and test when the remaining total is divisible by the y coefficient.