Problem
Four fours: hit a target number
Exactly four 4s may be used. Only operation signs and parentheses may be added. The four expressions must differ from one another. Write four expressions that all equal 8.
Reasoning · Grade 5-1 Number Puzzles
Your answer
Watch out 4 + 4 is tempting, but all four 4s must be used.
How to solve
Strategy Solve an Easier Related Problem — Four 4s at once is too much to search in my head, so I shrink the problem: first I ask the easier question "what numbers can I make with only TWO 4s?" and stock up a short list. Then I work backwards from the target: 8 is 12 - 4, 8 is 4 + 4, 8 is 8 ÷ 1, 8 is 32 ÷ 4 — each of these is a pair of numbers, and I only have to build each half out of 4s. Splitting four 4s into two halves of two 4s each is the move that makes the whole puzzle easy.
1STEP 1
Stock up: what can two 4s make?
Two 4s make 0, 1, 8, 16.
4 + 4 = 8, 4 - 4 = 0, 4 × 4 = 16, 4 ÷ 4 = 1, 44
Five values is a list short enough to keep in your head, and every four-4s expression that splits down the middle is just two of these values glued together with one more operation.
5.OA.A.2Make A Systematic List2STEP 2
Work backwards from 8
Routes to 8: 12 − 4, 4 + 4, 8 ÷ 1, 32 ÷ 4.
12 - 4 = 8, 4 + 4 = 8, 8 ÷ 1 = 8, 32 ÷ 4 = 8
Starting from the answer and walking backwards turns one hard search into several tiny searches, and a Grade 5 student already knows all the pairs of numbers that make 8.
5.OA.A.2Work BackwardsWorking backwards from 8 asks what pair of values would reach it, which turns building into a short search.
Why?
Every way of reaching 8 can be run in reverse, so asking what must have come before is as legal as building forwards.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Why?
The expression is its two halves joined by one operation, so naming the target for each half finishes the design.
🧱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
First expression: from 12 − 4 = 8
The first is 4 + 4 + 4 − 4.
4 + 4 + 4 - 4 = 12 - 4 = 8
This is the plainest one: repeated addition of 4 is skip-counting, and 4, 8, 12 then back down one step lands on 8.
5.OA.A.1Solve An Easier Related Problem4STEP 4
Second expression: from 4 + 4 = 8
The second is 4 × 4 ÷ 4 + 4.
4 × 4 ÷ 4 + 4 = 16 ÷ 4 + 4 = 4 + 4 = 8
Multiplying by 4 and then dividing by 4 undoes itself, so three 4s can quietly stand in for a single 4 — a trick worth remembering for every four-fours target.
5.OA.A.1Solve An Easier Related Problem5STEP 5
Third expression: from 8 ÷ 1 = 8
The third is (4 + 4) ÷ (4 ÷ 4).
(4 + 4) ÷ (4 ÷ 4) = 8 ÷ 1 = 8
Dividing by 1 changes nothing, so any expression worth 1 can be tacked onto an 8 for free. This is exactly the pattern the sample answers used for the target 1, run in reverse.
5.OA.A.1Guess And Check6STEP 6
Fourth expression: from 32 ÷ 4 = 8
The fourth is (4 + 4) × 4 ÷ 4.
(4 + 4) × 4 ÷ 4 = 8 × 4 ÷ 4 = 32 ÷ 4 = 8
Multiplying by 4 and dividing by 4 cancel, so this is the mirror image of the second expression: there the pair of 4s cancelled inside a product, here it cancels around an 8.
5.OA.A.1Guess And Check7STEP 7
Check all four
All four use four 4s and equal 8.
4+4+4-4 = 8, 4 × 4 ÷ 4 + 4 = 8, (4+4) ÷ (4 ÷ 4) = 8, (4+4) × 4 ÷ 4 = 8
Counting the 4s is the check students most often skip, and it is the one place this puzzle goes wrong — an expression worth 8 with only three 4s does not count.
5.OA.A.1Make A Systematic ListAnswer
4 + 4 + 4 − 4, 4 × 4 ÷ 4 + 4, (4 + 4) ÷ (4 ÷ 4), (4 + 4) × 4 ÷ 4
Every expression was evaluated with the order of operations and every one lands exactly on 8, with exactly four 4s used and nothing else. The target 8 is easy to reach because 4 + 4 already equals it, which means the other two 4s only have to be neutralised — by adding then subtracting, by multiplying then dividing, or by making a 1 to divide by. That is why so many answers exist, and it matches the sample answers for 1, which used the same neutralising idea.
Takeaway
Split four 4s into two pairs, find out what a pair can be worth, and the puzzle shrinks to "which two numbers make 8?"
- Stock up: what can two 4s make?
- Work backwards from 8
- First expression: from 12 − 4 = 8
- Second expression: from 4 + 4 = 8
- Third expression: from 8 ÷ 1 = 8
- Fourth expression: from 32 ÷ 4 = 8
- Check all four