Problem
Complete a 4 by 4 magic square
A 4 by 4 table already holds seven numbers. The numbers 1 to 16 are each used once. Every row, every column and both diagonals must have the same total. Fill in all the empty cells.
Reasoning · Grade 5-2 Varieties of Magic Squares
Your answer
Watch out Guessing the total wastes time: divide the grand total by four.
How to solve
Strategy Identify Subproblems — Nine blanks at once is far too much to hold in your head, but a single line with only one blank left is a one-subtraction subproblem: the missing number is the common total minus the three you can see. So I first work out the common total, then hunt each time for the line that has the fewest blanks, fill it, and look again — every number I write turns some other line into a one-blank line. Only one place needs more care, a line with two blanks, and there I list the number pairs that fit and knock out the ones that break another line.
1STEP 1
Work out the common total before placing anything
The total 136 makes each line 34.
1+2+3+…+16 = 8 × 17 = 136, 136 ÷ 4 = 34
Splitting the puzzle into 'first find the total, then fill the cells' turns a nine-blank mystery into a target number you can check against. Adding 1 to 16 by pairing and then sharing 136 equally among 4 rows is Grade 4 four-operation work.
4.OA.A.3Identify Subproblems2STEP 2
Fill the line that already has three numbers
The line with three numbers gives 2.
15+12+5 = 32, 34-32 = 2
A line with exactly one hole is the easiest possible subproblem: one subtraction and it is done. Always spend your first move on the fullest line, never on an empty one.
3.NBT.A.2Identify SubproblemsA line with exactly one hole is the easiest possible subproblem, because one subtraction finishes it.
Why?
The line total is its four numbers added together, so knowing the total and three of them leaves one part to recover.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
With only one candidate value that fits, no guessing is involved and the cell is forced.
🧱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.
3STEP 3
Column 1 now pins down a pair of numbers
Column 1 narrows to 13 and 16.
34-(3+2) = 29; 13+16 = 29, 14+15 = 29
When a line has two holes you cannot finish it in one step, but you can still list the handful of number pairs that reach the target. With numbers only up to 16, a total as big as 29 leaves just two pairs to write down.
3.NBT.A.2Make A Systematic List4STEP 4
Decide which of 13 and 16 goes on top
The diagonal puts 13 on top.
16+8+12 = 36 ＞ 34
With only two candidates left you do not have to guess — you test one, watch it overshoot the target of 34, and the other one wins. Comparing 36 with 34 is all the arithmetic it takes.
3.NBT.A.2Eliminate Possibilities5STEP 5
Row 3 has one hole left
Row 3's hole is 9.
16+5+4 = 25, 34-25 = 9
Each number written turns some other line into a one-hole line. Filling 16 into column 1 is exactly what made row 3 solvable in a single subtraction.
3.NBT.A.2Identify Subproblems6STEP 6
Now the other diagonal falls
The other diagonal gives 14.
3+8+9 = 20, 34-20 = 14
The diagonals are easy to forget, but they are lines just like the rows and columns, and here the diagonal is the only line with a single hole left — so it is the one to use next.
3.NBT.A.2Identify Subproblems7STEP 7
Finish row 2, and column 4 checks itself
Finishing row 2 makes column 4 check itself.
13+8+12 = 33, 34-33 = 1; 15+1+4+14 = 34
A line that fills up on its own and lands on 34 without being forced is a free check that nothing has gone wrong so far.
3.NBT.A.2Identify Subproblems8STEP 8
Column 3, then the last two cells
Fill column 3 and the last two cells.
34-(12+9+7) = 6; 34-(2+7+14) = 11; 34-(3+6+15) = 10
By this point every remaining line has exactly one hole, so the square finishes itself one subtraction at a time — no guessing anywhere.
3.NBT.A.2Identify Subproblems9STEP 9
Check all ten lines and all sixteen numbers
All ten lines total 34.
Ten line totals plus one sorted list is a complete check. Because every step was a forced subtraction rather than a guess, this square is the only one that fits the seven printed numbers.
4.OA.A.3Make A Systematic ListAnswer
3 10 6 15 / 13 8 12 1 / 16 5 9 4 / 2 11 7 14
136 ÷ 4 = 34
The common total has to be 34 and nothing else: the sixteen numbers add to 136 and the four rows share that equally, so 136 / 4 = 34. That is also the right size — four numbers averaging 8.5, the middle of 1 to 16, gives 4 x 8.5 = 34. All ten lines were re-added and every one comes to 34, and the sixteen entries sort to exactly 1 through 16 with no number used twice. Every blank was forced by a subtraction, and the one moment of choice (13 or 16 in column 1) was settled by 16 + 8 + 12 = 36 already overshooting 34, so this filling is the only one possible.
Takeaway
Find the one total every line must reach, then always fill the line that has just one empty cell — the square finishes itself, no guessing needed.
- Work out the common total before placing anything
- Fill the line that already has three numbers
- Column 1 now pins down a pair of numbers
- Decide which of 13 and 16 goes on top
- Row 3 has one hole left
- Now the other diagonal falls
- Finish row 2, and column 4 checks itself
- Column 3, then the last two cells
- Check all ten lines and all sixteen numbers