Guess My Number! From Binary Tricks to Bases and Beyond

We revisit the classic “guess my number” game and extend it from its familiar binary form to representations in any integer base. For each base we derive formulas for the number of cards needed to identify a given integer and, conversely, for the largest integer that can be determined when the number of cards is fixed. Both analysis and graphical evidence show that base 2 is optimal: it requires the fewest cards for any given number and, for a fixed card count, allows the widest range of integers to be guessed. Complete proofs are provided in the Appendix.