Magic Trick

Mathematical Principles & Theorems

Applies binary positional decomposition \(N = \sum_{j=0}^{k-1} b_j 2^j\) to information reconstruction. Each card \(j\) lists all numbers whose \(j\)-th bit is 1. Asking whether the secret number appears on card \(j\) extracts exactly 1 bit of information (halving entropy \(H(X)\) by Shannon's theorem: \(H = -\log_2(1/2) = 1\text{ bit}\)). With \(k\) cards, any number in \([1, 2^k - 1]\) is uniquely identified by summing the top-left indices \(2^j\) of positive cards.

Operating Instructions

• Mentally select a secret integer within the indicated range (1 to 63).
• Review each card sequentially and click 'Yes' if your number is present or 'No' if absent.
• Watch the binary accumulator reconstruct the exact binary bit pattern.
• Inspect the mathematical explanation card detailing how bitwise positional values compose the secret number.