I’m doing something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on various lessons I’ve learned while trying to answer the questions posed by gifted elementary school students. (This is updated from my previous index.)
Part 1: A surprising pattern in some consecutive perfect squares.
Part 2: Calculating 2 to a very large exponent.
Part 3a: Calculating 2 to an even larger exponent.
Part 3b: An analysis of just how large this number actually is.
Part 4a: The chance of winning at BINGO in only four turns.
Part 4b: Pedagogical thoughts on one step of the calculation.
Part 4c: A complicated follow-up question.
Part 5a: Exponentiation is multiplication as multiplication is to addition. So, multiplication is to addition as addition is to what? (I offered the answer of incrementation, but it was rejected: addition requires two inputs, while incrementation only requires one.)
Part 5b: Why there is no binary operation that completes the above analogy.
Part 5c: Knuth’s up-arrow notation for writing very big numbers.
Part 5d: Graham’s number, reputed to be the largest number ever to appear in a mathematical proof.
Part 6a: Calculating $(255/256)^x$.
Part 6b: Solving $(255/256)^x = 1/2$ without a calculator.
Part 7a: Estimating the size of a 1000-pound hailstone.
Part 7b: Estimating the size a 1000-pound hailstone.
Part 8a: Statement of an usually triangle summing problem.
Part 8b: Solution using binomial coefficients.
Part 8c: Rearranging the series.
Part 8d: Reindexing to further rearrange the series.
Part 8e: Rewriting using binomial coefficients again.
Part 8f: Finally obtaining the numerical answer.
Part 8g: Extracting the square root of the answer by hand.