Problem
Insert plus and minus signs to make 100
The three chains are 98○7○6○5○4○3○2○1, 9○8○76○5○4○3○21 and 9○8○7○65○4○32○1. Every circle takes a plus or a minus. Fill them so each chain equals exactly 100.
Reasoning · Grade 5-1 Making a Target Number
Your answer
Watch out A minus seems to cost the number, but flipping it from plus costs twice as much.
How to solve
Strategy Change Focus / Count the Complement — Trying signs at random is hopeless — chain (1) alone has 2⁷ = 128 sign patterns. So I change what I am looking for: first add everything with plus signs, then ask how far over 100 that lands, and finally ask which numbers to flip to minus. The key pattern is that flipping one number from + to - drops the total by twice that number, so the question turns into a tidy one — pick numbers whose sum is half the overshoot. That is a short list I can write out completely.
1STEP 1
Add every chain up with all plus signs
With all pluses every chain gives 126.
98+7+6+5+4+3+2+1 = 126, 9+8+76+5+4+3+21 = 126, 9+8+7+65+4+32+1 = 126
Computing the easiest case first, the all-plus one, turns a search problem into a measurement problem: how far away am I, and in which direction?
4.NBT.B.4Change Focus Count The Complement2STEP 2
See what one minus sign really costs
Flipping to minus costs twice that number.
(… + n + …) → (… - n + …): total drops by 2 × n
Noticing the doubling is the whole trick: it converts "choose plus or minus for seven circles" into "choose a handful of numbers with a particular sum".
5.OA.A.2Look For A PatternTurning one plus into a minus lowers the all-plus total by exactly twice that number.
Why?
Subtracting a number reverses adding it, so first the addition has to be undone and then the number taken away again.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Why?
Each term contributes its own amount to the total, so flipping one sign changes only that term's contribution.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
3STEP 3
Work out the exact target for the flipped numbers
So the flipped numbers must add to 13.
126-100 = 26, 26 ÷ 2 = 13
Halving the overshoot is the one computation that unlocks all three parts at once, and it is just a subtraction followed by a division by 2.
4.OA.A.3Change Focus Count The Complement4STEP 4
Chain (1): find numbers after the 98 that add to 13
In chain (1), 7 + 4 + 2 makes 13.
7+4+2 = 13 → 98-7+6+5-4+3-2+1 = 100
With only seven small numbers to choose from, a short organised sweep finds a group summing to 13 quickly, and the sum can be checked in one line.
4.NBT.B.4Make A Systematic List5STEP 5
Chain (1): check the answer and note there are others
That pattern gives 100.
98-7+6+5-4+3-2+1 = 100
Walking the finished chain left to right is an independent check that does not reuse the shortcut, so it would catch a slip in the halving step.
4.NBT.B.4Make A Systematic List6STEP 6
Chain (2): only one group adds to 13
In chain (2) only 8 + 5 works.
8+5 = 13 → 9-8+76-5+4+3+21 = 100
Throwing out the numbers that are already bigger than 13 shrinks the search to four small numbers, which can be checked completely in a few seconds.
4.NBT.B.4Make A Systematic List7STEP 7
Chain (3): again only one group adds to 13
In chain (3) only 8 + 4 + 1 works.
8+4+1 = 13 → 9-8+7+65-4+32-1 = 100
Once the oversized numbers are set aside there are only four numbers left, so listing every group and testing its sum is quick and leaves no gaps.
4.NBT.B.4Make A Systematic List8STEP 8
Check the two long chains by direct computation
Computing directly, all three give 100.
9-8+76-5+4+3+21 = 100, 9-8+7+65-4+32-1 = 100
Recomputing straight through, without using the 126-and-13 shortcut at all, is a genuinely independent check on both answers.
4.NBT.B.4Look For A PatternAnswer
7 + 4 + 2, 8 + 5, 8 + 4 + 1
126 − 100 = 26, 26 ÷ 2 = 13
Each chain was checked twice: once by the shortcut — all-plus total 126, overshoot 26, so flip numbers summing to 26 ÷ 2 = 13 — and once by adding the finished chain straight through from left to right, which gave exactly 100 every time. The magnitudes make sense too: 100 is a little under the all-plus total of 126, so only a small handful of small numbers should turn negative, and indeed the flipped numbers sum to just 13 in every chain. The overshoot 26 is even, which it must be, since every flip changes the total by an even amount.
Takeaway
Add everything first, see how far past 100 you land, then flip the signs of numbers adding to half that overshoot — because every minus sign takes away double the number.
- Add every chain up with all plus signs
- See what one minus sign really costs
- Work out the exact target for the flipped numbers
- Chain (1): find numbers after the 98 that add to 13
- Chain (1): check the answer and note there are others
- Chain (2): only one group adds to 13
- Chain (3): again only one group adds to 13
- Check the two long chains by direct computation