标题： 猜我的数字！ 从二进制技巧到一般的基数表示，需要多少张卡片？ 显示英文标题

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

摘要： 我们重新审视经典的“猜我的数字”游戏，并将其从熟悉的二进制形式扩展到任何整数基数的表示形式。 对于每个基数，我们推导出识别给定整数所需的卡片数量的公式，并反过来，当卡片数量固定时，推导出可以确定的最大整数的公式。 分析和图形证据表明，基数2在两个方向上都是最优的：它需要最少的卡片来表示任何指定的整数，并且在卡片数量固定时，允许猜测最广泛的整数范围。 图表说明了这些结果，完整的证明出现在附录中。 显示英文摘要

摘要： 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.