Mental arithmetic that helps in Chain

You do not need to be fast at arithmetic to solve Chain, but being fast changes what the puzzle feels like. If a multiplication costs you fifteen seconds, you will only try the branches you already believe in. If it costs you two seconds, you will try six branches and let the numbers tell you which one is right. The techniques below are the ones that actually pay off on this board — not general mental-maths tricks, but the specific shapes Chain keeps handing you. Every example is a real daily puzzle. If you have not read the rules yet, read those first; this page assumes you know what a tile does.

Double, then double again

Chain's multipliers are single digits, 2 to 9, and the running value is usually two or three digits. That is exactly the range where repeated doubling beats long multiplication.

×4 is two doublings. ×8 is three. On Friday 4 September the hard puzzle is start 21, target 216, hand +70 ×8 −78 +60 +59 swap, and the chain ends on ×8 applied to 27. Do not reach for a times table: 27, 54, 108, 216. Three halves of a second, and the last doubling is the one that tells you the answer is right.

The odd multipliers each have a shortcut worth learning:

• ×5 is ×10 then halve. On Wednesday 9 September the start is 14 and the chain opens ×5: 140, halved, is 70. That is faster than 14 × 5 for most people, and much faster once the running value is three digits.
• ×9 is ×10 minus the number itself. On Thursday 10 September the chain reaches 39 and plays ×9: 390 − 39 = 351. Subtracting 39 from 390 is easier than nine lots of anything.
• ×6 is double, then triple — or split it. On Saturday 5 September the chain opens ×6 on 52: doubling gives 104 and tripling gives 312. Splitting works just as well: 50 × 6 = 300, plus 2 × 6 = 12, so 312.
• ×3 is double and add the original. 39 × 3 = 78 + 39 = 117.

Division runs the same way backwards. ÷2, ÷4 and ÷8 are one, two and three halvings. On that same Saturday puzzle the hand also holds ÷4, and 52 ÷ 4 is 26 then 13 — perfectly legal, and a decoy. Being able to compute it in a second is what lets you dismiss it in a second.

Add by rounding, then pay the change back

Almost every +n and −n tile in Chain is a two-digit number, and most of them are close to a round one. Round to the ten or the hundred, do the easy sum, then correct.

• Monday 7 September has −28 in the winning set. 188 − 28 is 188 − 30 + 2 = 160.
• Tuesday 8 September opens +56 on 97. Take three from the 56 to finish the 97: 100 + 53 = 153. Later the same chain plays −96 on 612: 612 − 100 + 4 = 516.
• The Friday puzzle plays −78 on 150: 150 − 80 + 2 = 72.
• Sunday 6 September plays −37 on 186: 186 − 40 + 3 = 149.
• Wednesday 9 September finishes +79 on 99, which is two roundings at once: 100 + 80 − 1 − 1 = 178.

The pattern is always the same — move to a round number, then hand back the difference. What makes it worth the habit is that the intermediate is a number you can hold. 186 − 40 leaves 146, which you will not confuse with anything; 186 − 37 done digit by digit leaves you juggling a borrow.

Run the last digit alongside the value

The units digit of the running value follows its own small arithmetic, and it is cheap enough to carry in parallel with the real number. It costs almost nothing and it catches mistakes immediately.

Take Sunday 6 September: start 59, target 298, and the winning chain is +93 +34 −37 ×2. Run only the last digits: 9 + 3 = 12, so 2. 2 + 4 = 6. 6 − 7 borrows to 9. 9 × 2 = 18, so 8. The target ends in 8, so the chain is at least consistent. The real values are 152, 186, 149, 298 — and each one ends where the shortcut said it would.

The same on Tuesday 8 September: 7 + 6 = 13, so 3; 3 × 4 = 12, so 2; 2 − 6 borrows to 6. Target 516 ends in 6. The values are 153, 612, 516.

This is not the same use of last digits as the one in the strategy guide. There, last digits narrow down which tiles belong in the set before you play anything. Here, they are a running checksum on arithmetic you have already committed to. Both are worth doing, and they catch different mistakes.

Divisibility without dividing

A ÷n tile is refused unless the division is exact, so before you plan a chain around one you want to know whether the value will be divisible when you get there. The classical rules cover every divisor Chain uses except 7:

• 2, 4, 8: last digit even; last two digits divisible by 4; last three divisible by 8.
• 3 and 9: add the digits. 261 sums to 9, so it takes ÷9 — and 261 ÷ 9 = 29. 39 sums to 12, which is divisible by 3 but not 9, so ÷3 is legal there and ÷9 is not.
• 5: last digit 0 or 5.
• 6: even and digit sum divisible by 3.
• 7: no cheap rule that beats just dividing. Take the nearest multiple of 7 you know and count from there.

Thursday 10 September makes the digit-sum rule do real work. Start 36, target 117, hand ÷3 ×9 −28 +28 +31. The value reaches 39 (digit sum 12, divisible by 3) and from there both ÷3 ×9 and ×9 ÷3 land on 117 — because 351 sums to 9 and is divisible by 3 as well. On Saturday 5 September the same rule rules a move out: 312 sums to 6, so it takes ÷3 but not ÷9, and the chain has to reach 261 before the ÷9 tile will play.

What swap digits actually does to a number

Swap looks like a wildcard. It is not; it moves the value by a very predictable amount.

For a two-digit number, swapping changes the value by 9 × (units − tens). Thursday 3 September starts on 59 and the chain opens with a swap: 9 × (9 − 5) = 36, so 59 + 36 = 95. For a three-digit number it is 99 × (units − hundreds), because the middle digit does not move. The Wednesday 2 September chain swaps 134: 99 × (4 − 1) = 297, and 134 + 297 = 431. Going down works the same way — 72 swaps to 27, which is 9 × (2 − 7) = −45.

Two consequences are worth carrying around. First, the size of the jump is set entirely by the gap between the first and last digit, so 116 is a big swap (99 × 5 = 495, and 116 becomes 611) while 132 is a small one (99 × 1 = 99, and 132 becomes 231). Second — and this is the useful one — reversing digits never changes the digit sum, so a swap never changes whether the value is divisible by 3 or by 9. If you need a ÷3 later and the value is not divisible by 3 now, the swap tile will not rescue you.

Holding one number in your head

The reason Chain is playable without paper is that there is only ever one number to hold. There is no expression building up, no bracket waiting to close, no history you need except the current value — and the board shows you that anyway, with the chain written out in the margin. Your working memory is doing one job.

Three habits keep it that way. Plan in tiles, not in values: decide "multiply, then the two big additions, then subtract" and let the board compute. Land on round numbers before a multiplication when you have the choice, because 150 × 8 is a different experience from 147 × 8. And when you are unsure, play the tile instead of imagining it — undo is free, it costs no try, and a value you can read is worth more than a value you are holding.

The one thing worth doing on paper is the last-digit table on hard days, and even that is three lines. The FAQ covers the practical side — when the puzzle changes, what the timer counts, and what the site keeps on your device. If you find a shortcut we have missed, the feedback button on this page reaches us directly.