Maths Olympiad · topic 25 of 33 · free

Number puzzles and digit counting — Maths Olympiad practice

Digit puzzles look long but become quick when you count in groups: the ones place, then the tens place, or pages with 1, 2 and 3 digits. Magic squares hide a neat shortcut too — every row is one third of the total.

How to solve it

Count place by place — “split the count into ones, tens and hundreds”

  1. Count a digit place by place: first in the ones place, then in the tens place (a number like 33 counts twice).
  2. Page numbers: pages 1–9 use 1 digit each, 10–99 use 2, 100–999 use 3.
  3. A 3 × 3 magic square made of nine numbers: each row adds up to (sum of all nine) ÷ 3.

Use it for: Digit counting, page numbering, magic square and digit-sum puzzles in Class 5–8 Olympiad papers.

Worked example: A book has 120 pages, numbered 1, 2, 3 and so on. How many digits are printed to number all the pages? → 252

  1. Pages 1–9: 9 × 1 = 9 digits. Pages 10–99: 90 × 2 = 180 digits.
  2. Pages 100–120: 21 × 3 = 63 digits. Total: 9 + 180 + 63 = 252.

Check: If every page used 3 digits: 120 × 3 = 360; pages 1–9 save 2 each (18) and pages 10–99 save 1 each (90): 360 − 18 − 90 = 252 ✓

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