Puzzle for September 29, 2023 ( )
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
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Hint #1
Add A and F to both sides of eq.4: E - A + A + F = B - F + A + F which becomes eq.4a) E + F = B + A Add D and E to both sides of eq.3: F - D + D + E = B - E + D + E which becomes F + E = B + D which may be written as eq.3a) E + F = B + D
Hint #2
In eq.3a, replace E + F with B + A (from eq.4a): B + A = B + D Subtract B from each side of the equation above: B + A - B = B + D - B which makes A = D
Hint #3
In eq.5, replace C with B + E (from eq.2): B + E + F = A + B + D Subtract B from both sides of the equation above: B + E + F - B = A + B + D - B which becomes eq.5a) E + F = A + D
Hint #4
In eq.4a, replace E + F with A + D (from eq.5a): A + D = B + A Subtract A from each side of the above equation: A + D - A = B + A - A which makes D = B
Hint #5
Substitute D for B in eq.3a: E + F = D + D which becomes eq.3b) E + F = 2×D Subtract F from both sides of eq.3b: E + F - F = 2×D - F which becomes eq.3c) E = 2×D - F
Hint #6
Substitute D for B, and 2×D - F for E (from eq.3c) in eq.2: C = D + 2×D - F which becomes eq.2a) C = 3×D - F
Hint #7
Substitute D for A and B, 3×D - F for C (from eq.2a), and 2×D for E + F (from eq.3b) in eq.1: D + D + 3×D - F + D + 2×D = 34 which becomes 8×D - F = 34 In the above equation, add F to both sides, and subtract 34 from both sides: 8×D - F + F - 34 = 34 + F - 34 which becomes eq.1a) 8×D - 34 = F
Hint #8
To make eq.1a true, check several possible values for D and F: If D = 5, then F = 8×5 - 34 = 40 - 34 = 6 If D = 6, then F = 8×6 - 34 = 48 - 34 = 14 If D > 6, then F > 14 If D = 4, then F = 8×4 - 34 = 32 - 34 = -2 If D < 4, then F < -2 Since F must be a one-digit non-negative integer, the above equations make: F = 6 and D = 5
Hint #9
Since D = 5, then: A = D = 5 B = D = 5
Hint #10
Substitute 5 for D, and 6 for F in eq.3c: E = 2×5 - 6 which becomes E = 10 - 6 which makes E = 4
Solution
Substitute 5 for D, and 6 for F in eq.2a: C = 3×5 - 6 which becomes C = 15 - 6 which makes C = 9 and makes ABCDEF = 559546