Puzzle for April 8, 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
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Hint #1
In eq.2, replace D with A + C (from eq.3): A + E = C + A + C which becomes A + E = A + 2×C Subtract A from both sides of the above equation: A + E - A = A + 2×C - A which makes E = 2×C
Hint #2
In eq.4, replace A with E + F (from eq.6): F - E = E + F - B + E which becomes F - E = 2×E + F - B In the above equation, subtract F from each side, and add E to each side: F - E - F + E = 2×E + F - B - F + E which becomes 0 = 3×E - B Add B to each side of the above equation: 0 + B = 3×E - B + B which makes B = 3×E Substitute (2×C) for E: B = 3×(2×C) which makes B = 6×C
Hint #3
Substitute 6×C for B, and 2×C for E in eq.5: 6×C + C = D - 2×C Add 2×C to both sides of the equation above: 6×C + C + 2×C = D - 2×C + 2×C which means 9×C = D
Hint #4
In eq.3, substitute 9×C for D: 9×C = A + C Subtract C from each side: 9×C - C = A + C - C which makes 8×C = A
Hint #5
Substitute 8×C for A, and 2×C for E into eq.6: 8×C = 2×C + F Subtract 2×C from each side: 8×C - 2×C = 2×C + F - 2×C which means 6×C = F
Solution
Substitute 8×C for A, 6×C for B and F, 9×C for D, and 2×C for E in eq.1: 8×C + 6×C + C + 9×C + 2×C + 6×C = 32 which simplifies to 32×C = 32 Divide both sides by 32: 32×C ÷ 32 = 32 ÷ 32 which means C = 1 making A = 8×C = 8 × 1 = 8 B = F = 6×C = 6 × 1 = 6 D = 9×C = 9 × 1 = 9 E = 2×C = 2 × 1 = 2 and ABCDEF = 861926