Problem
Arrange cards for the smallest valid result
Use the cards 1, 2, 3, 6, 8, 9 once each to build (three-digit) − (three-digit). The result is taken as positive. Find (1) the second-smallest result and (2) the smallest when both numbers are odd.
Reasoning · Grade 3-1 Completing Equations
Your answer
Watch out The smallest, 18, is the easy answer — but (1) asks for the second-smallest.
How to solve
Strategy Make a Systematic List — A small difference comes from two numbers that are close together. We make the hundreds digits consecutive, then adjust the tens and ones to fine-tune the gap. Listing the closest pairs in order lets us read off the smallest and second-smallest, and the odd-only rule simply limits which cards can be the ones digits.
1STEP 1
A small difference needs two close numbers
A small difference needs the two numbers close together.
3□□ - 2□□
Numbers that are neighbors on the number line have the tiniest gap between them.
2.NBT.A.4Identify SubproblemsA small difference needs the two numbers to sit as close together on the number line as the cards allow.
Why?
The leading digits decide most of each number's size, so the two numbers can only be close if their leading digits are neighbours.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
Why?
Ranking the candidate pairs by how far apart they sit chains into one order, so the closest pair is the one at the front of that ranking.
🧱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.
2STEP 2
Find the smallest difference
With 3 and 2 on top, 316 − 298 = 18 is the smallest.
316 - 298 = 18
Push the bigger number down and the smaller number up so they crowd together as tightly as the cards allow.
3.NBT.A.2Guess And Check3STEP 3
Find the second-smallest difference (Part 1 answer)
Next up is 318 − 296 = 22, with nothing between.
318 - 296 = 22
List the closest pairs in increasing gap; the first is 18, the very next is 22.
3.NBT.A.2Make A Systematic List4STEP 4
Part 2 setup: both numbers odd
Ending odd means the ones digits are two of 1, 3, 9.
ones digits ∈ {1, 3, 9}
Only the last digit decides odd or even, so the puzzle is really about which two odd cards sit in the ones places.
2.OA.C.3Identify Subproblems5STEP 5
Find the smallest odd-odd difference (Part 2 answer)
The closest such pair is 921 − 863 = 58.
921 - 863 = 58
Even with the odd-ending rule, the trick is the same: make the two numbers neighbors so their difference is tiny.
3.NBT.A.2Guess And CheckAnswer
(1) 22, (2) 58
All results are positive and use each card once. The smallest possible difference is 18, and 22 is the next value up, so 22 is correctly the second-smallest. For the odd case, both 921 and 863 end in odd digits and 921 - 863 = 58 checks out as the minimum.
Takeaway
To make a subtraction result tiny, make the two numbers neighbors - and remember a number is odd only when it ends in an odd digit!
- A small difference needs two close numbers
- Find the smallest difference
- Find the second-smallest difference (Part 1 answer)
- Part 2 setup: both numbers odd
- Find the smallest odd-odd difference (Part 2 answer)