Mathematical Puzzles — Fundamental Concepts

Fundamental Concepts

For a two-digit number, use 10a + b, not a + b. Its reversal is 10b + a, so their difference is 9(a − b). If the digit sum is eleven and reversal makes the number twenty-seven smaller, solve a + b = 11 and a − b = 3 to obtain 74. Check the direction of the change as well as the digit sum.

Age puzzles add the same elapsed time to each person's age. If a parent is three times a child's present age and twice the child's age twelve years later, 3x + 12 = 2(x + 12) gives the child twelve. The age difference stays constant, while the ratio changes.

Transfers conserve the total and may describe alternative scenarios. With boxes initially holding 42 and 30 counters, moving six from A to B gives 36 each; instead moving six from B to A gives 48 and 24. Do not apply the second alternative to the already altered first state.

Remainder puzzles require all divisibility conditions and any size restriction. Values 32 + 35k leave remainder two modulo five and four modulo seven; the least above 100 is 102. For sequences, follow the supplied rule and test every transition. Finite terms alone can fit multiple rules, so distinguish an intended simple pattern from a mathematically unique continuation.