Mathematical Puzzles — Explained
Detailed Explanation
Turn the verbal puzzle into a mathematical model
A mathematical puzzle hides a small system of relationships inside a story. The difficulty often lies in deciding what the numbers represent, rather than in doing advanced calculation. Start by naming the unknown quantities, recording their units and translating each condition into an equation, inequality or divisibility requirement.
A candidate answer must satisfy the original wording as well as the algebra. Ages, counts and digits impose different restrictions, so a numerical solution is not automatically an admissible one.
Digit puzzles require place value
Consider an original puzzle: a two-digit number has digits whose sum is eleven, and reversing its digits makes it 27 smaller. Let the tens digit be a and the units digit b. The number is 10a + b, while its reversal is 10b + a. Their difference is 9(a − b), so a − b = 3. Together with a + b = 11, this gives a = 7 and b = 4. The number is 74, and 74 − 47 = 27 checks the result.
The place-value expression is essential. Treating the difference of the two numbers as merely a − b would lose the factor nine. Reversing the subtraction would instead produce 47, which makes the reversed number larger, contradicting the wording. Check that a lies from one to nine and b from zero to nine; a leading zero would not describe a two-digit number. In a different puzzle, a valid algebraic solution could still fail these digit constraints.
Age relationships change with time
Suppose a parent's present age is three times a child's, and twelve years later the parent will be twice the child's age. Let the child's age be x. The equation is 3x + 12 = 2(x + 12), giving x = 12 and the parent 36. After twelve years, their ages are 24 and 48, which have the required ratio. Adding twelve only to the parent would distort the passage of time. Both ages increase by the same number of years, while their difference remains constant.
Ratios generally change as people age. The present ratio three-to-one cannot simply be reused twelve years later. The invariant is the age difference, here 24 years. This distinction between a changing ratio and a constant difference is useful in many puzzles involving cumulative quantities. State whether ages are being treated as exact year values or completed years if the wording makes that relevant; an examination model normally supplies the intended precision.
Transfer puzzles conserve the total
Two boxes contain counters. If six counters move from box A to box B, the boxes contain equal numbers. If, instead, six counters move from B to A, A contains twice as many as B. Let their initial counts be a and b. The first condition gives a − 6 = b + 6, so a − b = 12. The second gives a + 6 = 2(b − 6). Substituting a = b + 12 produces b = 30 and a = 42.
Check the two separate experiments from the original state: moving six from A to B gives 36 and 36; moving six the other way gives 48 and 24. The total remains 72 in both. A frequent error is to apply the second transfer after the first, although the word instead signals an alternative scenario. Another is to subtract six from one box without adding it to the other, violating conservation. Labelling the initial state prevents both mistakes.
Remainders impose several conditions at once
Suppose the least positive integer leaves remainder two when divided by five and remainder four when divided by seven. Numbers satisfying the first condition are 2, 7, 12, 17, 22, 27, 32 and so on. Testing them against the second condition identifies 32, because 32 = 4 × 7 + 4. There is no need to invoke a sophisticated theorem for this small search. The answer must satisfy both remainders, and the request for the least positive value requires checking that earlier candidates fail.
If the puzzle additionally requires the number to exceed 100, 32 is no longer sufficient. Since the combined pattern repeats every 35, candidates are 32 + 35k, and the least above 100 is 102. A restriction in the final clause can change the answer even after the central congruences are solved. Keep every condition visible until the end.
A sequence needs a stated or justified rule
Short numerical sequences can fit many mathematical rules. In a standard aptitude exercise, choose the simplest rule supported by the displayed structure and answer options, but recognise that finite data do not determine a unique continuation without assumptions. For 3, 8, 15, 24, the successive differences are 5, 7 and 9; continuing the odd-number difference pattern gives 35. This is a reasonable intended rule, not a proof that no other generating formula could fit the four given values.
Explain the rule rather than asserting that a number feels natural. If alternating positions follow separate patterns, write the two subsequences. If the rule involves multiplication and addition, test it at every transition. A shortcut that works for one pair but fails for the others is not a valid explanation.
Verify the model as well as the answer
Use substitution, units, integer restrictions and a quick estimate to audit the result. A negative number of counters signals an inadmissible solution; a ratio equation with mixed units signals a modelling error. All examples here are original reasoning practice. Their purpose is to develop translation and verification skills for objective CSAT questions, not to claim a particular formula or puzzle has a guaranteed examination frequency.