Puzzle for September 19, 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.6, replace B with A + E (from eq.3), and D with E + F (from eq.2): A + F = A + E + E + F + E which becomes A + F = A + 3×E + F Subtract A and F from each side of the equation above: A + F - A - F = A + 3×E + F - A - F which simplifies to 0 = 3×E which means 0 = E
Hint #2
In eq.2, substitute 0 for E: D = 0 + F which makes D = F
Hint #3
In eq.3, substitute 0 for E: B = A + 0 which makes B = A
Hint #4
In eq.5, substitute 0 for E: D + 0 = B + C - 0 which becomes eq.5a) D = B + C
Hint #5
Substitute B + C for D (from eq.5a) into eq.4: C + B + C = A + B which becomes 2×C + B = A + B Subtract B from both sides of the above equation: 2×C + B - B = A + B - B which makes 2×C = A and also makes 2×C = A = B
Hint #6
Substitute 2×C for B in eq.5a: D = 2×C + C which makes D = 3×C and also makes D = F = 3×C
Solution
Substitute 2×C for A and B, 3×C for D and F, and 0 for E in eq.1: 2×C + 2×C + C + 3×C + 0 + 3×C = 11 which simplifies to 11×C = 11 Divide both sides of the above equation by 11: 11×C ÷ 11 = 11 ÷ 11 which means C = 1 making A = B = 2×C = 2 × 1 = 2 D = F = 3×C = 3 × 1 = 3 and ABCDEF = 221303