Prime Sieve Game

How to Play Prime Sieve

Use the sieve of Eratosthenes to find prime numbers in the displayed interval. A prime is a whole number greater than one whose only positive divisors are one and itself. The number 1 is excluded at the beginning because it is neither prime nor composite. Start with the displayed prime base, then cross out its remaining multiples.

For base 2, begin at 4 and mark the uncrossed multiples of 2. Check sieve step verifies that you have marked every required number before allowing the next base. For base 3, begin at 9; numbers such as 6 were already excluded during the earlier step. A number that is not a required multiple is rejected, with its current divisibility rule explained.

Continue until every prime base whose square is at most the upper limit has been processed. Crossed numbers have an × and a strike-through. Established primes have a ✓. After the last base, Confirm primes records the completed sieve. The remaining uncrossed numbers are exactly the primes; you are not asked to memorize a hidden list or react quickly.

Worked Sieve Through 20

For numbers through 20, the first step crosses out 4, 6, 8, 10, 12, 14, 16, 18 and 20 as multiples of 2. The next step uses base 3. Its multiples from 9 are 9, 12, 15 and 18, but only 9 and 15 still need marking. The marked numbers have a factorization that proves they are composite.

The next possible prime base is 5, and 5² = 25 is greater than 20. No further base is needed. The remaining primes are 2, 3, 5, 7, 11, 13, 17 and 19. A composite number has at least one prime factor no greater than its square root. Therefore checking the prime bases up to that bound excludes every composite in the interval.

Starting each step at the square of its base avoids repeated work. Smaller multiples of a prime already have a smaller prime factor and were marked during earlier steps. The game still shows those crossed numbers, with an accessible factor explanation. It never asks you to cross out the prime base itself: a prime number is not a composite multiple to eliminate.

Prime Number Practice and Saved Progress

Easy uses a short interval up to 20, 25 or 30. Normal extends the upper limit to 40, 50 or 60. Hard uses intervals up to 80, 90 or 100 and includes additional sieve bases where needed. These are difficulty settings for one mathematical task. Larger grids remain playable with scrolling and compact, readable touch buttons.

Tap a number or reach it with Tab and activate it using Enter or Space. Undo restores the previous marking or checked step and marks assistance. A hint gives one justified factorization or tells you that the current step is ready to check. Daily mode changes at midnight UTC; Practice mode lets you request another interval and saved task.

Correct marking steps do not incur a time penalty. Incorrect choices reduce the score by five points, and each hint reduces it by fifteen. Completed scores have a floor of ten. Unaided bests are stored by difficulty, and each distinct puzzle counts once. Your current base and crossed numbers resume locally when saving is available; blocked storage still permits play for the current visit.

Frequently Asked Questions

Why is 1 excluded?

1 is neither prime nor composite. Prime numbers have exactly two positive divisors.

Why start crossing at the prime’s square?

Smaller composite multiples already have a smaller prime factor and were excluded during earlier sieve steps.

When can I stop sieving?

After every prime base whose square is at most the upper limit has been processed, the remaining numbers are prime.