Puzzle for December 7, 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
In eq.2, replace B with A + C (from eq.4): F = A + A + C which becomes eq.2a) F = 2×A + C
Hint #2
In eq.3, replace F with 2×A + C (from eq.2a): D = A + 2×A + C which becomes eq.3a) D = 3×A + C
Hint #3
In eq.5, substitute 3×A + C for D (from eq.3a), and 2×A + C for F (from eq.2a): C + 3×A + C = E + 2×A + C which becomes 2×C + 3×A = E + 2×A + C Subtract 2×A and C from both sides of the equation above: 2×C + 3×A - 2×A - C = E + 2×A + C - 2×A - C which becomes C + A = E which is the same as eq.5a) A + C = E
Hint #4
Substitute E for A + C (from eq.5a) in eq.4: B = E
Hint #5
Substitute (A + C) for B (from eq.4), 3×A + C for D (from eq.3a), 2×A + C for F (from eq.2a), and A + C for E (from eq.5a) in eq.6: A - (A + C) + 3×A + C + 2×A + C = C + A + C - A which becomes A - A - C + 5×A + 2×C = 2×C which becomes 5×A + C = 2×C Subtract C from each side of the equation above: 5×A + C - C = 2×C - C which makes 5×A = C
Hint #6
Substitute 5×A for C in eq.4: B = A + 5×A which makes B = 6×A and also makes B = E = 6×A
Hint #7
Substitute 5×A for C in eq.3a: D = 3×A + 5×A which makes D = 8×A
Hint #8
Substitute 5×A for C in eq.2a: F = 2×A + 5×A which makes F = 7×A
Solution
Substitute 6×A for B and E, 5×A for C, 8×A for D, and 7×A for F in eq.1: A + 6×A + 5×A + 8×A + 6×A + 7×A = 33 which simplifies to 33×A = 33 Divide both sides of the above equation by 33: 33×A ÷ 33 = 33 ÷ 33 which means A = 1 making B = E = 6×A = 6 × 1 = 6 C = 5×A = 5 × 1 = 5 D = 8×A = 8 × 1 = 8 F = 7×A = 7 × 1 = 7 and ABCDEF = 165867