Problem
Four fours: reach several large targets
Six numbers must be built: 15, 20, 44, 60, 68 and 80. Each one uses exactly four 4s. Only operation signs and parentheses may be added — no other digits. Write an expression for each of the six.
Reasoning · Grade 5-1 Number Puzzles
Your answer
Watch out The 4s look separate, but two may be joined into 44.
How to solve
Strategy Make a Systematic List — Guessing whole expressions out of the air is hopeless. Instead I first make a short, complete list of every number two 4s can make: 4-4=0, 4÷4=1, 4+4=8, 4×4=16, and the run-together number 44. That tiny list is the whole toolbox. Then I split each target into two pieces — one piece built from two 4s and one from the other two — which turns each target into a small subproblem I can finish by trying the few candidates on the list. Because the targets here are large, the piece that carries most of the size is usually 4×4×4=64, 4×4+4=20, or the two-digit 44.
1STEP 1
List everything two 4s can make
Two 4s make 0, 1, 8, 16, 44.
4-4=0, 4÷4=1, 4+4=8, 4×4=16, 44
Two 4s and one sign is a small enough world to list completely — five entries — and a complete list means I can never be surprised later.
5.OA.A.2Make A Systematic ListListing everything two 4s can make gives a short menu, and every four-4s expression is two of those joined by one sign.
Why?
Two 4s and one operation is a small closed world, so writing it out completely leaves nothing that could turn up later.
🧱Disjoint cases addSplit the possibilities so no case overlaps another and none is left out, and the case counts simply add.
Why?
The left half and the right half can be chosen without either limiting the other, so every pairing from the menu really is available.
🧱Independent choices multiplyIf a first choice can go a ways and each of those leaves b ways for a second, the pair can be made a times b ways.
2STEP 2
List the useful values three 4s can make
Three 4s usefully make 64, 20, 12, 11.
4×4×4=64, 4×4+4=20, 4×4-4=12, 44÷4=11
Grade 4 multiplication of whole numbers is all this needs; 4×4×4 is just 16×4, which a fourth grader can do in one step.
4.NBT.B.5Make A Systematic List3STEP 3
Target 15: land on 16 and step down by 1
15 is 16 minus 1: 4 × 4 − 4 ÷ 4.
16-1=15 → 4×4-4÷4=16-1=15
Splitting 15 as 16-1 is the whole idea: each half of the split is something I already know how to build with two 4s.
5.OA.A.1Identify Subproblems4STEP 4
Target 20: build 5 and multiply by 4
20 is 5 times 4: (4 ÷ 4 + 4) × 4.
5×4=20 → (4÷4+4)×4=5×4=20
Once you see 20=5×4, only the 5 is left to build, and 5 is just 1+4 — the 1 being the standard 4÷4 trick.
5.OA.A.1Guess And Check5STEP 5
Target 44: use the two-digit 44 itself
44 uses the two-digit form: 44 ÷ 4 × 4.
11×4=44 → 44÷4×4=11×4=44
Dividing by 4 and then multiplying by 4 undoes itself, so those two 4s act like a 'do nothing' pair — a very handy move whenever the target is already on the board.
5.OA.A.1Identify Subproblems6STEP 6
Targets 60 and 68: sit on 64 and step 4 either way
60 and 68 step 4 either way from 64.
64-4=60 → 4×4×4-4=60; 64+4=68 → 4×4×4+4=68
Two targets for the price of one: once 64 is in reach, every number within 4 of it is one sign away.
4.OA.A.3Identify Subproblems7STEP 7
Target 80: build 20 and multiply by 4
80 is 20 times 4: (4 × 4 + 4) × 4.
20×4=80 → (4×4+4)×4=20×4=80
The same split-into-two-pieces habit works for the biggest target too; only the size of the pieces changes.
5.OA.A.1Guess And Check8STEP 8
Count the 4s in every answer
Each expression uses exactly four 4s.
16-1=15; 5×4=20; 11×4=44; 64-4=60; 64+4=68; 20×4=80
Miscounting the 4s is the one mistake that ruins an otherwise correct expression, so counting them is worth a separate pass.
5.OA.A.1Make A Systematic ListAnswer
4 × 4 − 4 ÷ 4, (4 ÷ 4 + 4) × 4, 44 ÷ 4 × 4, 4 × 4 × 4 − 4, 4 × 4 × 4 + 4, (4 × 4 + 4) × 4
Every expression uses the digit 4 exactly four times and no other digit, and recomputing each one gives its target: 16-1=15, 5x4=20, 11x4=44, 64-4=60, 64+4=68, 20x4=80. The sizes make sense too: the small target 15 needed only 4x4=16 as its backbone, the middle targets 44 and 60-68 needed a three-4 value (44 or 64), and the largest target 80 needed the biggest backbone of all, 20x4. I also checked exhaustively that 44 is unreachable from four separate 4s using only +, -, x, /, which is exactly why the run-together 44 appears in that one answer and nowhere else.
Takeaway
List what just two 4s can make first — 0, 1, 8, 16 and 44 — and every big target becomes 'one of those, adjusted by the other two 4s'.
- List everything two 4s can make
- List the useful values three 4s can make
- Target 15: land on 16 and step down by 1
- Target 20: build 5 and multiply by 4
- Target 44: use the two-digit 44 itself
- Targets 60 and 68: sit on 64 and step 4 either way
- Target 80: build 20 and multiply by 4
- Count the 4s in every answer