Balance a magic square
A normal 3 × 3 magic square uses each number from 1 to 9 exactly once. Every row, every column and both main diagonals must have the same sum. This is more restrictive than simply making the rows balance.
The numbers total 45. Since the three rows partition all nine entries and have equal totals, each row must sum to 15. The same target applies to columns and diagonals. The classic Lo Shu arrangement below meets those conditions.
Why must the center be 5?
Try an argument that works for every normal 3 × 3 magic square, not just the arrangement shown.
Hint: count the four lines through the center
Add the middle row, middle column and both diagonals. Every outside entry is counted once. The center is counted four times.
Solution: compare two totals
The four lines total 4 × 15 = 60. Counting by entries gives 45 + 3c, where c is the center: the original total already includes it once. Thus 45 + 3c = 60, so c = 5. Each opposite pair on a line through the center must then sum to 10.
The 24 game: fractions can be useful
For the version here, use four supplied numbers exactly once each, together with addition, subtraction, multiplication, division and parentheses, to make 24. Do not concatenate digits, use factorials or insert extra numbers. Division by zero is not allowed.
Try 3, 3, 8, 8. Whole-number intermediate results are not required. That detail matters: restricting yourself to integers would hide a valid route.
Hint: make a small denominator
Since 8 divided by 1/3 is 24, investigate whether the other three numbers can produce 1/3.
Solution: preserve the fraction exactly
8 ÷ (3 − 8 ÷ 3) = 8 ÷ (1/3) = 24
Both 8s and both 3s appear exactly once. Keep 8/3 as a fraction while checking: a rounded decimal can make the equality look approximate when it is exact.
The reusable idea is working backward from the target. You can ask what factor, difference or denominator would be helpful, then see whether the remaining numbers can make it. This narrows the search without promising that every four-number set is solvable.
Nim: examine what you leave behind
In normal-play Nim, two players alternate turns. On a turn, remove one or more counters from exactly one heap. The player who takes the last counter wins. The “last counter loses” version is called misère play and needs different endgame treatment; do not mix the two rules.
For normal play, the key calculation is the nim-sum: the bitwise exclusive OR, or XOR, of the heap sizes. In each binary position, XOR records whether the number of 1s is odd. It adds without carrying.
Take heaps of 1, 3 and 5. In binary, these are 001, 011, 101. Their XOR is 111, or 7, so the position has a winning move under perfect play. Reduce the heap of 5 to 2. The remaining sizes 1, 3 and 2 have XOR zero.
Why is zero important? Any legal move from a zero nim-sum changes at least one bit and leaves a nonzero nim-sum. From a nonzero nim-sum, there is a legal reduction restoring zero: use a heap with a 1 in the highest nonzero bit of the current nim-sum. Repeating this response ultimately leaves your opponent without a move.
Charles L. Bouton published the mathematical analysis of Nim in 1901. You can play with beans or marks on paper before learning binary notation. Small examples make the strategic distinction visible without turning the game into a speed test.
The strongest move often depends on the position you leave, not the number you take.
Look for the shared structure
Magic squares organize equal sums. The 24 game explores equivalent expressions. Nim tracks a property of a changing position. Another useful lens is modular arithmetic: numbers can behave alike when they leave the same remainder after division. This is the idea behind familiar clock arithmetic and many divisibility tests.
Albrecht Dürer included a 4 × 4 magic square in his engraving Melencolia I. Its scale and arrangement differ from Lo Shu, but it invites the same habit: check the claimed sums rather than trusting the label.
For more mathematical play, visit the University of Edinburgh Maths Circle. If a rule or step still feels unclear, find ways to use a hint without losing the investigation.