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Learn by Doing: Number Sense

Fall 2026 · 12 problems

Magnitudes, digits and decimals, worked by hand: how many digits a sum of powers up to a googolplex has, the pattern hiding in 1/9899, and iterated digit sums. Then six competition problems in the same vein.

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What is in it

Problems only — the club works these out together at meetings. Topic and attribution are listed as a starting point.

1. Unsimplified answers don't count for much! Algebra / Numerical AnswersEasy
2. Powers of (Almost) Equal Number Sense / EstimationMedium
3. From hundred, to googol, to googolplex Number Sense / Magnitudes
4. Sum of digits of sum of digits of sum of digits of… Number Sense / Digits
5. Integer sequences in rationals! Number Sense / Periodicity
6. Upstream, Downstream Number / Algebra SenseEasy
7. The Little Grasshopper that Could Number Sense / OptimizationMediumPutnam 2021 A1
8. Every Composite, Almost a Product Number Theory / AlgebraEasyPutnam 1988 B1
9. Two, Three, Four Number Theory / DiophantineMediumPutnam 2024 A1
10. d-d-death (to 2016) Number Theory / FactorialsMediumPutnam 2016 A1
11. \(A^3 + B^3 + C^3 - 3ABC =\)? Algebra / FactoringEasyPutnam 2019 A1
12. What's up, \(\mathbb{F}_{1979}\)? Number Theory / PairingHardIMO 1979, Problem 1