I think you would get better guesses in your system by using 6 instead of 7. An even-digit number lies in the range [10^(n-1), 10^n), e.g. when a number has 4 digits, it is at least 1,000 and less than 10,000. The geometric mean of these extremes is 10^((2n - 1)/2), The square root of which is 10^((2n-1)/4) = 10^(n/2) * 10^(-1/4) When n is odd, you get a power of 10 times 1.778... , which is close to 2 but when n is even, you get a power of 10 times 5.623... , which is much closer to 6 than to 7