Guess my number! From binary tricks to general base representations, how many cards are needed?

Abstract: 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 in both directions: it requires the fewest cards to represent any specified integer and, for a fixed card count, allows the widest range of integers to be guessed. Figures illustrate these results, and complete proofs appear in the Appendix.