Problem
Build an equation meeting given conditions
A staircase sum: one-digit + two-digit + three-digit = four-digit. Each digit 0 to 9 is used exactly once overall. No number may lead with 0. Find the largest and the smallest possible sum.
Reasoning · Grade 3-1 Completing Equations
Your answer
Watch out Piling the big digits into the addends is tempting, but the total needs digits too.
How to solve
Strategy Make a Systematic List — Think about place value: each digit contributes by where it sits. To make the sum large we want big digits in high-value addend places; to make it small we want small digits there. Then list candidate fillings near those targets and check the addition is correct and uses each digit once.
1STEP 1
See how each place pulls on the sum
The hundreds slot weighs most and the ones slots least.
addend total = 100h + 10(t₁ + t₂) + (o₁ + o₂ + o₃)
A digit matters more when it sits in a bigger place, so the hundreds slot is the strongest lever on the sum.
2.NBT.A.1Identify SubproblemsA digit does far more work in the hundreds place than in the ones, so the hundreds slot is the strongest lever on the sum.
Why?
The same digit standing in the hundreds place counts for a hundred of itself but only once in the ones place.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
Why?
The sum is what all the place amounts add up to, so raising the heaviest place lifts the total more than any other change could.
🧱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
Aim high for the largest sum
Big digits high gives the maximum 3 + 75 + 984 = 1062.
3 + 75 + 984 = 1062
We cannot freely pick the biggest digits because the four left-over digits have to BE the answer, so we list the strong candidates and keep the legal ones.
2.NBT.B.7Make A Systematic List3STEP 3
Confirm the largest sum uses every digit once
Counting the digits used: 0 to 9, once each.
984 + 75 + 3 = 1062
The check makes sure no digit is repeated or missing, which is the real difficulty of the puzzle.
2.NBT.B.7Guess And Check4STEP 4
Aim low for the smallest sum
The total must reach four digits, so the minimum is 3 + 45 + 978 = 1026.
3 + 45 + 978 = 1026
The sum can't be smaller than 1000 (it has 4 digits), so we hunt for the lowest 4-digit number that the leftover digits can actually form.
2.NBT.B.7Make A Systematic List5STEP 5
Confirm the smallest sum uses every digit once
Its digits do not repeat either, so both stand.
978 + 45 + 3 = 1026
Verifying the digit set confirms this small total is really allowed, not just a number we wished for.
2.NBT.B.7Guess And CheckAnswer
largest 1062, smallest 1026
Both sums are 4-digit numbers just above 1000, which makes sense because the most three addends like these can reach is around 1065 and they must total at least 1000. Both example fillings add up correctly and use each of 0-9 exactly once.
Takeaway
Knowing that hundreds count for more than ones lets you steer an addition to its biggest or smallest answer - that's just Grade 2 place-value sense!
- See how each place pulls on the sum
- Aim high for the largest sum
- Confirm the largest sum uses every digit once
- Aim low for the smallest sum
- Confirm the smallest sum uses every digit once