Lesson Guide
Explanation
In a magic square every row, every column and both diagonals add to the same total. So the first job is always to find a line that is already complete, because that line hands you the total. A top row of eight, one and six gives fifteen, and from then on every other row, column and diagonal must also make fifteen. Without that total you cannot start; with it, most of the square falls out one cell at a time.
Your Turn
A complete row reads 2, 7, 6. What is the magic total? Check: 15.
Watch Out
The total comes from a COMPLETE line. A row with a gap in it tells you nothing yet.
Explanation
Once you have the total, look for a line with exactly one empty cell and fill it. Never guess at a line with two gaps — there is always another line with only one. With a total of fifteen, a column holding eight and three needs four at the bottom. Every cell you fill opens another line, so keep circling the square until it is complete, and check the diagonals at the end because they are the ones people forget.
Your Turn
The total is 15 and a row reads 4, 9 and a gap. What goes in the gap? Check: 2.
Watch Out
Check the diagonals too. A square can look right by rows and columns and still fail a diagonal.
Explanation
A sequence where each term is the sum of the three before it runs easily once you have three starting numbers. One, two and three give six; two, three and six give eleven; three, six and eleven give twenty. Always test the rule on a term you already have before using it on one you do not — six plus eleven plus twenty is thirty-seven, which matches, so the rule holds and the next term is sixty-eight.
Your Turn
The sequence runs 1, 2, 3, 6, 11, 20, 37. What comes next? Check: 11 + 20 + 37 = 68.
Watch Out
Add the three terms IMMEDIATELY before. Skipping one or including the term itself gives a different sequence.
Explanation
When the unknown sits in the middle of the sequence, find the equation with only one unknown in it. Here the fourth term, fifteen, is made from two, five and the missing term, so the missing term is fifteen minus seven, eight. Then check it forwards on the next term: five plus eight plus fifteen is twenty-eight, which matches. Two lines agreeing is what tells you the answer is right.
Your Turn
In 2, 5, ?, 15, 28, find the missing term. Check: 15 − 2 − 5 = 8.
Watch Out
Use a term you KNOW as the total. Building an equation out of two unknowns gets you nowhere.
Explanation
Both puzzles use the same move: find the line with exactly one unknown and fill it, then look again. In the square, the middle column holds one and five with a gap, and with a total of fifteen that gap is nine; the nine then opens the bottom row, and the bottom row opens the last corner. In the sequence it is the same — one equation with one unknown, solved, then used. Quiz and exercises now.
Your Turn
The magic total is 15 and the middle column reads 1, 5, ?. What is the gap? Check: 9.
Watch Out
After each fill, look again from the start. A cell you just wrote often unlocks two more lines.
📌 Key Points — this lesson in a nutshell
- Find the one line you can complete first — a full row of a magic square, or three terms in a row of a sequence.
- In a magic square every row, column and diagonal has the same total. One complete line gives you that total.
- For a sequence with a rule like "each term is the sum of the previous three", work forwards where you can and backwards where you must.
- Backwards through a sum: if a + b + c = d and you know three of them, the fourth is d minus the others.
Step 1: Introduction
Step 1 of 5
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