Puzzle for December 13, 2019 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
Scratchpad
Help Area
Hint #1
Subtract B and C from both sides of eq.4: B + D – B – C = A + C – B – C which becomes D – C = A – B In eq.3, replace A – B with D – C: D – E = D – C Subtract D from each side of the above equation: D – E – D = D – C – D which makes –E = –C which means E = C
Hint #2
In eq.2, replace E with C: C + D = C + F Subtract C from both sides of the equation above: C + D – C = C + F – C which makes D = F
Hint #3
In eq.6, substitute C for E, and 40 for A + B + C + D + E + F (from eq.1): (C × C) – F = 40 which becomes C² – F = 40 In the equation above, add F to each side, and subtract 40 from both sides: C² – F + F – 40 = 40 + F – 40 which becomes eq.6a) C² – 40 = F
Hint #4
To make eq.6a true, check several possible values for C and F: If C = 9 then F = 9² – 40 = 81 – 40 = 41 If C = 8 then F = 8² – 40 = 64 – 40 = 24 If C = 7 then F = 7² – 40 = 49 – 40 = 9 If C = 6 then F = 6² – 40 = 36 – 40 = –4 If C < 6 then F < –4 Since F must be a non-negative one-digit integer, then F = 9 which makes C = 7 and also makes D = F = 9 and E = C = 7
Hint #5
Substitute 9 for F, and 7 for C and E in eq.5: A + 9 = 7 + 7 which becomes A + 9 = 14 Subtract 9 from each side of the equation above: A + 9 – 9 = 14 – 9 which makes A = 5
Solution
Substitute 9 for D, 5 for A, and 7 for C in eq.4: B + 9 = 5 + 7 which becomes B + 9 = 12 Subtract 9 from each side of the above equation: B + 9 – 9 = 12 – 9 which means B = 3 and makes ABCDEF = 537979