h coin numbers 2-26 balance out, the reference coin (number 1) cannot be fake, SO coin number 27 is the fake one. 3.4 Rényi's fake coin A B C Split A B C Split into A B C A = B select C A = B select C A = B select C A < B select A A < B select A A < B select A A > B select B A > B select B A > B select B Step 1 Step 2 Step 3 Figure 3.1 Solution of Rényi's fake-coin problem: the 27 coins are split into three groups (A, B, C). The scale first compares weights of groups (A, B), which identifies the group X = A, B, or C containing the fake (lighter) coin. The same process is repeated with the selected group, X being split into three groups of three coins (step 2), and three groups of one coin (step 3). The final selection is the fake coin. There exists at least one other way to solve Rényi's fake-coin determination problem, which to the best of my knowledge I believe is original. 11 The solution consists in making the three measurements only using groups of nine coins. The idea is to assign to each coin a position in space, forming a 3 x 3 coin cube, as illustrated in Fig. 3.2. The coins are thus identified or labeled with a coefficient Cijk, where each of the indices i, j, k indicates a plane to which the coin belongs (coin C111 is located at bottom-left on the front side and coin C333 is located at top-right on the back side). To describe the measurement algorithm, one needs to define nine groups corresponding to all possible planes. For the planes defined by index i = const., the three groups are: P1 = (C111, C112, C113, C121, C122, C123, C131, C132, C133) = {C1jk} Pi=2 = (C211,C212, C213, C221, C222, C223, C231, C232, C233) = {C2jk} (3.10) P=3 = (C311, C312, C313, C321, C322, C323, C331, C332, The other definitions concerning planes j = const. and = const. are straightforward. By convention, we mark with an asterisk any group containing the fake coin. If we put then P1 and Pi=2 on the scale (step 1), we get three possible measurement outcomes (the signs meaning group weights being equal, lower, or greater): Pi=1=