Puzzle for May 6, 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.
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Hint #1
In eq.2, replace F with B + C (from eq.3): D = E + B + C which may be written as eq.2a) D = B + E + C
Hint #2
In eq.2a, replace B + E with C (from eq.4): D = C + C which makes D = 2×C
Hint #3
In eq.1, substitute D for A + B (from eq.5), and D for E + F (from eq.2): D + C + D + D = 28 which becomes eq.1a) 3×D + C = 28
Hint #4
Substitute (2×C) for D in eq.1a: 3×(2×C) + C = 28 which becomes 6×C + C = 28 which makes 7×C = 28 Divide both sides of the above equation by 7: 7×C ÷ 7 = 28 ÷ 7 which makes C = 4 and also makes D = 2×C = 2 × 4 = 8
Hint #5
eq.1 may be written as: A + D + E + B + C + F = 28 In the equation above, replace B + C + F with A + D + E (from eq.6): A + D + E + A + D + E = 28 which becomes 2 × (A + D + E) = 28 Divide both sides by 2: 2 × (A + D + E) ÷ 2 = 28 ÷ 2 which becomes eq.1b) A + D + E = 14
Hint #6
In eq.6, replace A + D + E with 14 (from eq.1b): eq.6a) B + C + F = 14
Hint #7
Substitute 8 for D in eq.2a: 8 = B + E + C Subtract E from each side of the equation above: 8 - E = B + E + C - E which becomes eq.2b) 8 - E = B + C
Hint #8
Substitute 8 - E for B + C (from eq.2b) in eq.6a: 8 - E + F = 14 In the above equation, subtract 8 from both sides, and add E to both sides: 8 - E + F - 8 + E = 14 - 8 + E which becomes eq.6b) F = 6 + E
Hint #9
Substitute 8 for D, and 6 + E for F (from eq.6b) in eq.2: 8 = E + 6 + E which becomes 8 = 2×E + 6 Subtract 6 from each side of the equation above: 8 - 6 = 2×E + 6 - 6 which makes 2 = 2×E Divide both sides by 2: 2 ÷ 2 = 2×E ÷ 2 which makes 1 = E
Hint #10
Substitute 1 for E in eq.6b: F = 6 + 1 which makes F = 7
Hint #11
Substitute 7 for F, and 4 for C in eq.3: 7 = B + 4 Subtract 4 from each side of the above equation: 7 - 4 = B + 4 - 4 which makes 3 = B
Solution
Substitute 8 for D, and 3 for B in eq.5: 8 = A + 3 Subtract 3 from both sides of the equation above: 8 - 3 = A + 3 - 3 which makes 5 = A and makes ABCDEF = 534817