Down adds 10 (47), up takes away 10 (27), down-right adds 11 (48) and down-left adds 9 (46). The rules break at the edges, where the grid wraps round: 40 is not “one to the right” of 39 in the same way.

Even makes five columns (2, 4, 6, 8, 10). With Odd as well, all 100 are shaded and none are striped in two colours: every whole number is either odd or even, never both.

Multiples of 5 fill the 5 and 10 columns. The cells striped in both colours are the 10 column, because 10 = 2 × 5, so every multiple of 10 is also a multiple of 5. 15 is a multiple of 5 but not of 10.

19: ten with 7 in the units (7, 17, … 97) and ten in the seventies, but 77 is in both lists so it’s only counted once. Most classes predict 10 or 20.

The first column and last column are usually quickest. Listen for strategies: “it’s in the sixties row, 4 along, so 64” (watch the last column: 70 sits in the row with 61 to 69, not at the start of the seventies). Counting on cell by cell is the strategy to move pupils away from.

On the 1–100 grid every row ends with a multiple of 10, so 40 finishes the row that starts with 31. On a 0–99 grid every row is one decade (40 to 49), and the tens digit matches the row. Many primary teachers use 0–99 for exactly this reason.

Going down one and left one adds 9, which is a multiple of 3, so you stay on a multiple of 3. ×9 slopes the same way, ×11 slopes the other way (down-right adds 11), and ×6 makes every other ×3 diagonal.

The tens digit goes up by 1 and the units digit goes down by 1 each time, because adding 9 is adding 10 and taking away 1. There are 11 multiples of 9 up to 100. 9 to 81 make one diagonal, and 90 and 99 start the next one.

The multiples of 10. A number divisible by 2 and by 5 is divisible by 10, because 2 and 5 have no common factor.

The multiples of 12 (12, 24, … 96): 8 numbers. Many pupils predict multiples of 24, because 4 × 6 = 24. The lowest common multiple is 12, not 24, because 4 and 6 share the factor 2.

Again the multiples of 12, and this time 3 × 4 = 12. Multiplying works when the two numbers share no factor (3 and 4). It overshoots when they do (4 and 6, or 6 and 8).

26 are left. They are 1, 49, 77, 91 and the 22 primes above 5. So 1, 49 (7 × 7), 77 (7 × 11) and 91 (7 × 13) are the non-primes. “Not a multiple of 2, 3 or 5” is not the same as “prime”.

14, because 100 ÷ 7 = 14 remainder 2, and 7 × 14 = 98. Pupils who say 14.28 or 15 are a good discussion.

Remainder 1 is the ×3 pattern moved one place to the right (34 numbers), and remainder 2 is moved two places (33 numbers). With all three on, every cell is shaded once and nothing is striped: every whole number leaves remainder 0, 1 or 2 when you divide by 3.

24 has 8 factors. The most is 12, shared by 60, 72, 84, 90 and 96. Hover with ℹ Facts on to check factor counts quickly. Watch for pupils who miss the factor pair partner (finding 3 but not 8).

11 to 99 make a down-right diagonal, because down-right adds 11. 110 is still on that diagonal (it’s one down and right of 99), then 121 wraps round to the first column and a new diagonal starts. The ‘repeated digits’ pattern pupils notice stops at 99.

On 9 columns the multiples of 3 make three vertical columns (3, 6, 9). Going down one row now adds 9, which is a multiple of 3, so you stay in the same column.

On 15 columns they make three columns (5, 10 and 15). On 7 columns they make diagonals, because going down adds 7, which is not a multiple of 5.

On 9 columns, going down adds 9, so the multiples stack in the last column. On 8 columns, going down adds 8 and down-right adds 9, so they slope down to the right. On 10 columns, down-left adds 9, so they slope the other way.

Vertical: 7 or 14 columns (down adds a multiple of 7). Down-left adds (columns − 1), so 8 or 15 columns. Down-right adds (columns + 1), so 6 or 13 columns.

Any multiple of 4: 4, 8, 12, 16 or 20 columns. If the row length is a multiple of 4, going down a row adds a multiple of 4. 2 columns is a common wrong answer: it puts them in one column but only every other row.

12 is a multiple of both 3 and 4, so both sets of multiples make vertical columns: 3, 6, 9, 12 and 4, 8, 12. The 12 column has both colours because those are the common multiples.

The first and fifth columns: numbers one more or one less than a multiple of 6. The other columns hold multiples of 6, numbers that are even (6n + 2, 6n + 4), or multiples of 3 (6n + 3). Every prime above 3 is 6n ± 1, but not every 6n ± 1 is prime (25, 35, 49…).

A prime has exactly two factors: 1 and itself. 1 has only one factor, so it is not prime. It is not composite either. Many pupils (and some adults) think 1 is prime.

2 crosses out 49 numbers (4 to 100). 3 crosses out only 16 new ones: 32 multiples of 3 apart from 3, but 16 of those are even and already gone. The message under the grid gives the count.

Every multiple of 4 is a multiple of 2, so they are already crossed out. The same goes for 6, 8, 9 and 10. The sieve only ever uses primes.

11 × 11 = 121, which is beyond 100. Any multiple of 11 up to 100 (22, 33… 99) has a smaller factor too, so it is already gone. To sieve up to a number, you only need primes up to its square root.

Only 49, 77 and 91 (7 × 7, 7 × 11, 7 × 13). Every other multiple of 7 has a factor of 2, 3 or 5 and was crossed out earlier. The colour shows each number’s smallest prime factor.

13, because 13 × 13 = 169 is under 200 but 17 × 17 = 289 is not. There are 46 primes up to 200.

25 primes. All except 2 are odd, because every other even number has 2 as a factor. That makes 2 the only even prime.

Every number in that column ends in 4, so it is even. Above the first row, primes only appear in the 1, 3, 7 and 9 columns. The 2, 4, 6, 8 and 10 columns are even, and the 5 column is multiples of 5.

The biggest gap is from 89 to 97: seven numbers (90 to 96) with no prime. There are 8 twin prime pairs: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73). Nobody knows whether twin primes go on for ever: it is one of the big unsolved problems in maths.

There are 21 primes from 101 to 200, against 25 from 1 to 100. Primes thin out as numbers get bigger, but slowly, and they never stop.

The gaps are the odd numbers 3, 5, 7, 9… The gap after 100 is 21, so the next square is 121. The gap between n² and (n + 1)² is 2n + 1.

Squares only appear in the 1, 4, 5, 6, 9 and 10 columns, so a square number never ends in 2, 3, 7 or 8. You can use this to rule out 7,843 as a square straight away.

Two consecutive triangular numbers always add up to a square number (1 + 3 = 4, 3 + 6 = 9, 6 + 10 = 16). Two copies of a triangle of dots fit together to make a square.

1 and 36. There are infinitely many, but they get rare fast: the next is 1225 (35² and the 49th triangular number).

7 cubes: 1, 8, 27, 64, 125, 216 and 343. 1 and 64 are both square and cube (64 = 8² = 4³). The next one is 729 = 27² = 9³.

10 cells: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 (1 appears twice in the sequence but once on the grid). Each term is the sum of the two before, so each is about 1.6 times the last, and the gaps keep growing.

2, 8 and 34: every third Fibonacci number. The pattern goes odd, odd, even, because odd + odd = even, then odd + even = odd, then even + odd = odd.

2, 3, 5, 13 and 89. Nobody knows whether there are infinitely many Fibonacci primes: it is an open problem. Pupils sometimes guess every odd Fibonacci number is prime, but 21 and 55 are not.

Yes to all three: 1 = 1², 1 = 1³ and 1 is the first triangular number. Pupils often leave 1 out of lists of square numbers.

23 to 58 is down 3 rows (+30) and right 5 (+5), so +35. Going back is up 3 (−30) and left 5 (−5): −35. Pupils see addition and subtraction as the same jump in opposite directions.

Good mental maths practice: rows are tens, columns are ones. Listen for pupils who add the ones first, then the tens, and for anyone who moves the wrong way on a “left” jump.

Both routes add 36: +40 − 4, or −4 + 40. The order doesn’t change the total. Most pupils find “add 40 then take 4” easier than “+36” in one go: it is the compensation strategy.

Down-left always adds 9 (+10 − 1), and down-right always adds 11 (+10 + 1). To add 9, add 10 and take 1 away. It goes wrong at the edges of the grid.

55 below, 35 above, 44 left and 46 right. Up and down (±10) are usually harder than left and right (±1), and that is what the grid is good for practising. Click 45 again before each new jump so every jump starts from 45.

Yes: one clue is enough, because the layout fixes every other number. A second clue lets pupils check. A good extra clue is in the last column (like 60), because pupils find the “end of the row” hardest.

Now up adds 10, so “higher on the grid” means “bigger number”, like a graph or a building with floors. Both layouts are used in schools. It’s worth showing both so pupils think about the structure, not the position.

One row down adds 1, and one column across adds 0.1. After 0.9 comes 1 (one whole), not 0.10. Ask what 0.10 would mean. The grid shows 1.0 as “1” so that it lines up with 2, 3 and 4.

Yes. 0 = 2 × 0, so it is even, and 0 = 5 × 0, so it is a multiple of 5 (and of every number). Some books only count positive multiples, but 0 is definitely even.

×3 shades every cell. ×2 and ×6 shade half (the even multiples of 3: 50 cells). ×9 shades about a third (33). ×4 and ×12 shade a quarter (25: the multiples of 12). ×5 shades a fifth (20) and ×7 shades 14. Each new factor you need, beyond the 3, cuts the share down.

The column starting 5 (5, 25, 45…) and the column starting 15 (15, 35, 55…). Every row adds 20, so each of those columns is always a multiple of 5. Only 5 itself is prime.

Nothing is shaded on the first grid, because adding 2 to an even number keeps it even. On the second grid every cell is shaded. A step of 2 keeps the parity of the start number.

Odd and even keep alternating through zero (−3 is odd, −2 is even), and ×3 keeps its diagonals (−3, −6…). There are no negative square numbers on the grid, because squaring a negative gives a positive: (−4)² = 16.

Yes. The countdown grid is the 1–100 grid turned upside down (rotated 180°), and turning a diagonal through 180° keeps its slope. Down-left still adds a multiple of 9 (−10 + 1 = −9).

The rule is “3 more than a multiple of 4” (3, 7, 11, 15…): 25 numbers. 51 is shaded (48 + 3) and 99 is shaded (96 + 3). Pupils often say “the odd numbers in every other column”, which works on this grid but not in general.

The factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. The biggest shaded number is the clue: everything else divides into it. Pair them up (1 × 60, 2 × 30…) to check none is missing.

The rules include times tables, remainder rules, factors of a number, primes, squares, cubes and triangular numbers. A good habit to praise is testing a number that would prove the rule wrong, not just ones that fit.

Look for rules that are hard to spot from the first few numbers alone. For example, 1, 2, 4, 8, 16 then 31 (the maximum number of regions you get by joining points on a circle) shows why a pattern isn’t a proof.

90 can be done in 9 ways (7 + 83, 11 + 79, 17 + 73, 19 + 71, 23 + 67, 29 + 61, 31 + 59, 37 + 53, 43 + 47). 4, 6, 8 and 12 can each be done in only one way. Nobody has proved the conjecture, but it has been checked far beyond any number you could write down.

35 numbers (43 if you allow 0², which adds the squares themselves). The odd primes that work (5, 13, 17, 29…) all leave remainder 1 when divided by 4, and those that leave remainder 3 (3, 7, 11, 19…) never do. That is Fermat’s theorem on sums of two squares.

Strong investigations make a prediction first, then explain the pattern with the number of columns (what one row down adds). Shared links open the tool exactly as it was set up.