Problem
Solve using the sum of one line
The numbers 1 to 10 fill the 10 cells of an L-shaped figure, one each. A column of 6 and a row of 5 share one cell. The column totals 33 and the row totals 30. Find the number in the shared cell.
Reasoning · Grade 5-2 Foundations of Magic Squares
Your answer
Watch out 6 + 5 looks like 11 cells, but there are 10 — one is counted twice.
How to solve
Strategy Change Focus / Count the Complement — Trying to work out which numbers go where would mean testing a huge pile of arrangements, so I change focus and stop caring about individual cells: I only count the whole collection twice, once cell by cell and once line by line. The two outlines behave exactly like two overlapping loops of a Venn diagram whose overlap is the single cell A, so adding the two line totals counts every cell once except A, which gets counted twice. Comparing that with the total of 1 to 10 gives A straight away, with no arrangement needed at all.
1STEP 1
Count the cells and spot the double count
6 + 5 is 11 but there are 10 cells: one overlaps.
6 + 5 = 11, 11 - 10 = 1 cell counted twice
Counting the cells inside each outline and comparing with the cells actually drawn is simple counting, and the leftover 1 points straight at the corner cell that belongs to both lines.
2.OA.A.1Draw A Diagram2STEP 2
Add up all ten numbers
The numbers 1 to 10 total 55.
1+2+3+…+10 = 5 × 11 = 55
Pairing the smallest with the largest turns a long addition into five identical elevens, and because it uses every number once the answer 55 is fixed before anything is placed.
3.OA.A.3Look For A Pattern3STEP 3
Add the two line totals and see what you really counted
The two line totals add to 63.
33 + 30 = 63 = (all ten numbers) + A
Counting the same collection of numbers in two different ways is allowed to give two different-looking answers only if you can say exactly what the difference is — here it is one extra copy of A.
4.OA.A.3Change Focus Count The ComplementAdding the two line totals counts every cell once except the shared one, which is counted twice.
Why?
The shared cell lies on both lines, so it is counted once from each while every other cell belongs to one line only.
🧱Counted twice, so halveIf every item got counted from both of its two ends, the total is exactly double the truth, so halving fixes it.
Why?
The combined total is therefore all ten numbers plus one extra copy of the shared cell, so that extra copy can be peeled off.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
4STEP 4
Solve for A
The difference is the shared cell: 8.
55 + A = 63 → A = 63 - 55 = 8
Once the overlap is described in words, finding it is a single subtraction within 100 — no searching through arrangements is needed.
2.NBT.B.5Draw A Venn Diagram5STEP 5
Show a filling that really works
A real filling confirms it.
8+1+2+3+9+10 = 33, 8+4+5+6+7 = 30
Producing one actual filling proves the answer is possible and not just a number that survived the arithmetic.
4.OA.A.3Draw A DiagramAnswer
8
63 − 55 = 8
A has to be one of the numbers 1 to 10, and 8 is in that range. The size makes sense too: the red line has 6 numbers averaging 33 divided by 6, about 5.5 each, while the blue line has 5 numbers averaging 30 divided by 5, exactly 6 each, so a fairly large number like 8 sitting in both lines is no surprise. The check filling 8, 1, 2, 3, 9, 10 down the red column and 8, 4, 5, 6, 7 along the blue row uses each of 1 to 10 once and hits both 33 and 30 exactly.
Takeaway
Add the two line totals: the only square counted twice is the corner where the lines cross, so the extra is exactly the number hiding there.
- Count the cells and spot the double count
- Add up all ten numbers
- Add the two line totals and see what you really counted
- Solve for A
- Show a filling that really works