Problem
Recover hidden digits in a vertical sum
The column subtraction 9A5 − 7B9 = C56 must come out exactly. A, B and C are different single digits. Choose them so that A + B + C is as large as possible.
Reasoning · Grade 3-1 Completing Equations
Your answer
Watch out Reading the hundreds as 9 − 7 = 2 misses a case: if the tens borrows, C is 1 — and that case wins.
How to solve
Strategy Make a Systematic List — Work column by column like a real vertical subtraction, tracking borrows. The ones and hundreds columns are forced, which pins down C and splits the tens column into a small number of cases. Then we list those cases and pick the one with the biggest A + B + C.
1STEP 1
Ones column: a borrow is needed
5 cannot take 9, so borrowing gives 15 − 9 = 6.
15 - 9 = 6
This is just regrouping in subtraction: when the top digit is too small, you trade one ten for ten ones.
3.NBT.A.2Draw A DiagramWhen the top digit in the ones column is too small to subtract from, one ten is traded in for ten ones.
Why?
One ten is exactly ten ones, so moving it down changes only how the same amount is written, never how much it is.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
Why?
The number is what its place amounts come to together, so shifting an amount from the tens to the ones leaves the number itself untouched.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
2STEP 2
Hundreds column tells us C, in two cases
Depending on the tens borrow, C is 2 or 1.
9 - 7 = 2 or 9 - 1 - 7 = 1
Whether the middle column borrows decides what is left in the hundreds, so there are just two possible values of C.
3.NBT.A.2Make A Systematic List3STEP 3
Tens column in each case
That gives A − B = 6, or else B − A = 4.
Case 1: A - B = 6, C = 2 Case 2: B - A = 4, C = 1
Reading one column at a time turns the puzzle into two tidy little relationships between A and B.
3.NBT.A.2Make A Systematic List4STEP 4
Maximize A + B + C in each case
Maximised, the cases give 14 and 15 — the second wins.
Case 1 max = 14, Case 2 max = 15
To make a sum big you want the digits themselves big, so push A and B to their largest allowed values.
1.NBT.B.3Guess And Check5STEP 5
Check the winning arrangement
Checking: 955 − 799 = 156.
955 - 799 = 156
Plugging the numbers back in is the surest way to know the puzzle really balances.
3.NBT.A.2Guess And CheckAnswer
A = 5, B = 9, C = 1 (955 − 799 = 156, sum 15)
955 - 799 = 156 checks out, and 156 matches C56 with C = 1. The three digits 5, 9, 1 are distinct as required. The competing case gave only 14, so 15 is indeed the maximum.
Takeaway
If you read a subtraction one column at a time and keep track of borrowing, a 'hidden digit' puzzle is just Grade 3 subtraction you already know!
- Ones column: a borrow is needed
- Hundreds column tells us C, in two cases
- Tens column in each case
- Maximize A + B + C in each case
- Check the winning arrangement