Number Target Guide: Hit the Total Exactly

Look at the target before the numbers

Start by asking what the target is near. Is it close to a multiple of one of the big numbers? Is it a few away from a round hundred? Targets are three digits and the small numbers are 1 to 10, so almost every solution has the shape "a big product, adjusted by a small amount". Write the target as (something × something) ± (a small number) and look for the parts.

For a target of 437 with a 75 among the numbers: 75 × 6 = 450, and 450 − 437 = 13. Can you make 13 from what is left? With 10 and 3, yes. The whole puzzle takes fifteen seconds once you look at it this way.

The big number is a multiplier, not an addend

If the numbers include a 25, 50, 75 or 100, it is almost always multiplied, and the thing it is multiplied by is built from the small numbers. Divide the target by the big number and look at the quotient and remainder:

• 100 into 683 goes 6 remainder 83. Can you make 6 and 83? 83 is hard. Try 7 × 100 − 17 instead: 7 and 17 may be easier (17 = 10 + 7, or 2 × 8 + 1).
• 75 into 528 goes 7 remainder 3. 7 × 75 = 525, plus 3. Easy if a 3 or a 1 + 2 is available.
• 50 into 341 goes 6 remainder 41, or 7 with 9 short. 7 × 50 − 9 is usually the better route.

Try both the floor and the ceiling of the division; one of them normally leaves a remainder you can build.

Building a multiplier from small numbers

When the quotient you need is not among the small numbers, make it: 7 = 3 + 4, 12 = 3 × 4, 11 = 10 + 1, 9 = 10 − 1, 6 = 2 × 3. Keep one small number in reserve for the final adjustment. If you have 2, 3, 4 and 10 and need 7 × 75 + 3, neither 3 + 4 nor 10 − 3 will do for the 7, because both use the 3 you need at the end. (10 + 4) ÷ 2 = 7 uses the other three numbers and leaves the 3 free. This juggling is the real game: when the obvious parts collide, swap one for an equivalent.

Leaving remainders and the order of steps

The game builds the answer one step at a time: pick a number, an operation, another number, and they combine into a new number you can use again. So plan the steps so that every intermediate is a positive whole number. Division is only useful when it comes out exact: 75 ÷ 3 = 25 is fine; 75 ÷ 2 is not allowed. Subtraction must stay above zero: 3 − 10 is refused, so do 10 − 3 and remember you have the sign the other way.

Build the big product first, then adjust. If you build the adjustment first you may find you have used the number the product needed.

When the target is small or prime

Targets under 200 are often reachable without the big number at all, or by dividing: 100 ÷ 4 = 25, then 25 × 7 = 175. Prime targets cannot be a product, so they are always "product ± something": 347 = 350 − 3 = 7 × 50 − 3. Do not waste time trying to factor a target that ends in 1, 3, 7 or 9 unless you have seen a factor pair instantly.

Practice habits

• Learn the multiples of 25, 50 and 75 up to 10 times. 75 × 7 = 525 should be instant.
• Learn to see near-doubles and near-halves: 198 is 2 × 100 − 2; 450 is 9 × 50.
• Say the target's neighbourhood aloud: "a bit over 400", "just under 700". It frames the first multiplication.
• Use Undo rather than Reset when one step goes wrong; the earlier steps are usually fine.

Common mistakes

1. Adding when you should multiply. Six numbers rarely add to a three-digit target; multiplication does the heavy lifting.
2. Using the small numbers for the product and the big one for the adjustment. The other way round, almost always.
3. Forgetting intermediates must be whole and positive. Plan divisions and subtractions so they come out clean.
4. Fixing on the first shape. If the remainder will not build in twenty seconds, try the other side of the division or a different factor.
5. Using every number. You do not have to. A shorter solution is just as correct.

Glossary

• Target: the three-digit number to reach exactly.
• Intermediate: the result of one step, which becomes a new number you can use.
• Quotient and remainder: the whole-number result and what is left when you divide the target by a big number.
• Adjustment: the small number added or subtracted after the main product.