Problem
Move one matchstick to fix an equation
Two matchstick equations are wrong as printed. The digits are shaped like digital-clock numerals. Only one stick may be moved in each equation. Make both equations true.
Reasoning · Grade 4-2 Games of Wit
Your answer
Watch out Removing a stick looks easier, but a move keeps the count the same.
How to solve
Strategy Guess and Check — First do the arithmetic exactly as printed. Seeing how wrong the equation is says which number has to change and by how much, which is far more useful than randomly nudging sticks. Then, for the number that has to change, list every digit reachable by taking one stick out or putting one stick in, and check the short list against the arithmetic. Most branches die instantly because a stick has nowhere legal to land, so only a handful of candidates ever need real checking.
1STEP 1
Equation (1): do the subtraction as printed
As printed, (1) gives 29 — one too many.
53 - 24 = 29 ≠ 28
A two-digit subtraction with one regrouping is ordinary fourth-grade arithmetic, and knowing the gap is exactly 1 turns a vague puzzle into a precise target.
4.NBT.B.4Guess And Check2STEP 2
Equation (1): decide which number has to move
Leave the answer alone and fix the first number.
Comparing two digits bar by bar before touching anything kills whole branches at once, because a single move can change only one slot.
4.NBT.B.4Eliminate Possibilities3STEP 3
Equation (1): turn the 3 into a 2 with one slide
Turning the 3 into a 2 gives 52 − 24 = 28.
52 - 24 = 28
Laying the two digit shapes side by side and spotting the one bar that differs is looking at line segments in a figure, which fourth graders do all the time.
4.G.A.1Guess And Check4STEP 4
Equation (2): do the multiplication as printed
As printed, (2) gives 114 — far too big.
19 × 6 = 114 ≠ 70
Multiplying a two-digit number by a one-digit number is a fourth-grade skill, and here it immediately shows the mismatch is huge, not a near miss.
4.NBT.B.5Guess And Check5STEP 5
Equation (2): the answer cannot grow, so a factor must shrink
So a factor must shrink.
Comparing 114 against a two-digit answer box settles at once which side of the equation must give way.
4.NBT.B.5Eliminate PossibilitiesSince the answer cannot be made bigger by one stick, a factor has to shrink instead.
Why?
Making one factor smaller makes the product smaller, so the comparison between the printed product and the target fixes which side must move.
🧱Transitivity of orderIf the first is smaller than the second and the second smaller than the third, the first is smaller than the third, so pairwise comparisons chain into one full order.
Why?
Every way of changing the answer is ruled out by the stick count, so only the factors are left to change.
🧱Rule out the rest and the last one is forcedIf every possibility but one has been ruled out, the survivor must be the answer — no further checking is needed.
6STEP 6
Equation (2): list what 19 can become
The 19 can become 10, 13, 15 or 16.
19 → 10, 13, 15, 16
There are only six bars in a 9, so a young solver can rub out each one in turn and see which leftovers still read as digits.
4.G.A.1Make A Systematic List7STEP 7
Equation (2): match the loose stick to the answer
Placing the moved stick gives 13 × 6 = 78.
13 × 6 = 78
Once the possible left-hand numbers are down to two, multiplying each by 6 and looking at what the answer would have to become finishes the job.
4.NBT.B.5Guess And CheckAnswer
52 − 24 = 28, 13 × 6 = 78
3 → 2, 19 → 13
Check the arithmetic first: 52 minus 24 really is 28, and 13 times 6 really is 78. Then check the stick count, which a legal move must leave alone. Equation (1) starts with 5 + 5 + 5 + 4 + 5 + 7 = 31 sticks and finishes as 52 minus 24 equals 28 with 5 + 5 + 5 + 4 + 5 + 7 = 31, unchanged. Equation (2) starts with 2 + 6 + 6 + 3 + 6 = 23 sticks and finishes as 13 times 6 equals 78 with 2 + 5 + 6 + 3 + 7 = 23, unchanged. No sign was touched and every number still has the same count of digits, so both moves are legal. Checking every single-stick move by machine confirms each equation has exactly one solution, so nothing better was missed.
Takeaway
Do the sum as written first: how far off it is tells you which number has to change and by how much.
- Equation (1): do the subtraction as printed
- Equation (1): decide which number has to move
- Equation (1): turn the 3 into a 2 with one slide
- Equation (2): do the multiplication as printed
- Equation (2): the answer cannot grow, so a factor must shrink
- Equation (2): list what 19 can become
- Equation (2): match the loose stick to the answer