Logical Squares: Find the missing value! problem 1: 3 24 4 5 120 100 1 0 ? problem 2: ? 2 5 8 7 9 11 6 13 4 17 18 19 21 23 24 SPOILER Solution for problem 1: ----------------------- This problem is much too small to make a serious guess. Even the problem composer gives two alternative values. Solution: Relation between the first and the middle columns model 1: first, middle t t^3-t 0 0 1 0 2 6 3 24 4 60 5 120 6 210 7 336 model 2: first, middle t ((t^2+4t-1)/4)^2 - 1 1 0 3 24 5 120 7 360 Solution: Relation between the middle and the last columns model 1: middle, last t (t/12)^2 Therefore we get for t=0 the missing value ? = 0. model 2: middle, last t t-20 Therefore we get for t=0 the missing value ? = -20. Solution: Relation between the first and the last columns model 1: first, last t 48t - 140 1 -92 3 4 5 100 Therefore we get for t=1 the missing value ? = -92. model 2: first, last t (4t-10)^2 1 36 3 4 5 100 Therefore we get for t=1 the missing value ? = 36. Solution: linear dependence Determine '?' such that the three vectors become linear dependent. | 3 24 4 | | 5 120 100 | = 0 Therefore '?' = -8. | 1 0 -8 | This is the relation 48x - 7y + 6z = 0. Reference: R. Zehl; Denken mit Spass 2, Die neuesten trend-Nuesse, ORAC Verlag, Wien, 1984, ISBN 3-85368-983-3, Problem 11 Solution for problem 2: ----------------------- Read the numbers from left to right: ? 2 5 8 7 9 11 6 13 4 17 18 19 21 23 24 Drop every second number: ? 5 7 11 13 17 19 23 Solution 1: We have the sequence of odd primes: '?' = 3 Solution 2: We have the numbers 6n+-1 in ascending order: '?' = 1 Reference: New Beetle Riddle, Autohaus Menkel, Frankenberg (Eder), Germany