Pi=2 Pi=1 < Pi=2 Pit (3.11) P=1 > Pi=2 We then proceed with step two, taking, for instance, Pj=1 and Pj=2, which yields either 11 As proposed in 2004 by J.-P. Blondel of Alcatel (private discussion), which I have reformulated here in algorithmic form. Measuring information Figure 3.2 Alternative solution of Rényi's fake-coin problem: each of the 27 coins is assigned a coefficient Cijk, corresponding to its position within intersecting planes (i, j, k = 1, ,2,3) forming 13 x 3 cube. The coefficients C111 and C333, corresponding to the coins located in front-bottom-left and back-top-right sides (respectively) are explicitly shown. of the outcomes: Pj=1 = (3.12) Pj=1 > P == We then finally proceed with step three, taking, for instance, Pk=1 and Pk=2, which yields either of the outcomes: Pk=1 < Pk=2 (3.13) Pk=1 > Pk=2 -2. We observe that the three operations lead one to identify three different marked groups. For instance, assume that the three marked groups are , P- This immedi- ately tells us that the fake-coin coefficient cijk is of the form C3jk and Cilk and Cij2, or in mathematical notation, using the Kronecker symbol,12 Cijk Cijk8i38j18k2 C312. By definition,Sij=ji=friandj0 = otherwise. 3.5 Exercises ensemble-theory language, this solution is given by the ensemble intersection = P-2}. Note that this solution can be generalized with N = 3P coins where p is the dimension of a hyper-cube with three coins per side. The alternative solution to the Rényi fake-coin problem does not seem too practical to implement physically, considering the difficulty in individually labeling the coins, and the hassle of regrouping them successively into the proposed arrangements. (Does it really save time and simplify the experiment?) It is yet interesting to note that the problem accepts more than one mathematically optimal solution. The interest of the hyper-cube algorithm is its capacity for handling problems of the type N = m P where both m and p can have arbitrary large sizes and the measuring device is m-ary (or gives m possibl