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.

Figures (5)

• Figure 1: Top: Giuseppe Polone during a street performance in Naples (photo sourced from polonephoto). Bottom: the six binary tables (cards) used in the classic number-guessing game for integers from $1$ to $60$.
• Figure 2: Example of the game in base $3$: the six tables needed to guess any number from $1$ to $26$. Each pair of tables corresponds to one power of $3$, with separate tables for coefficients $1$ and $2$ of that power.
• Figure 3: Number of cards $N_{card}$ required to represent integers as a function of $n$, for bases $b=2,\dots ,10$. For each fixed base the stepwise growth reflects the addition of a new block of $(b-1)$ cards whenever $n$ passes a power of $b$.
• Figure 4: For integers $n=8$ to $16$, the number of cards $N_{card}$ required to guess $n$ as a function of the base $b$.
• Figure 5: Maximum representable integer as a function of the number of cards $N_{card}$ for bases $b=2,\dots ,10$.

Theorems & Definitions (4)

• Theorem 1
• proof
• Theorem 2
• proof