Puzzle for April 11, 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
eq.6 may be written as: A + B = C + D + F + E In the equation above, replace C + D + F with B (from eq.2): A + B = B + E Subtract B from each side of the equation above: A + B - B = B + E - B which makes A = E
Hint #2
In eq.2, replace D + F with A + C (from eq.3): B = C + A + C which becomes eq.2a) B = A + 2×C
Hint #3
In eq.4, substitute A + 2×C for B (from eq.2a): A + 2×C + C = A - C + D which becomes A + 3×C = A - C + D In the above equation, subtract A from both sides, and add C to both sides: A + 3×C - A + C = A - C + D - A + C which simplifies to 4×C = D
Hint #4
Substitute (A + 2×C) for B (from eq.2a), and A for E in eq.5: (A + 2×C) - F = A - (A + 2×C) + A which becomes A + 2×C - F = A - A - 2×C + A which becomes A + 2×C - F = -2×C + A In the above equation, subtract A from both sides, and add F and 2×C to both sides: A + 2×C - F - A + F + 2×C = -2×C + A - A + F + 2×C which simplifies to 4×C = F
Hint #5
Substitute 4×C for D and F in eq.3: 4×C + 4×C = A + C which becomes 8×C = A + C Subtract C from each side of the above equation: 8×C - C = A + C - C which makes 7×C = A and also makes 7×C = A = E
Hint #6
Substitute 7×C for A in eq.2a: B = 7×C + 2×C which makes B = 9×C
Solution
Substitute 7×C for A and E, 9×C for B, and 4×C for D and F in eq.1: 7×C + 9×C + C + 4×C + 7×C + 4×C = 32 which simplifies to 32×C = 32 Divide both sides of the above equation by 32: 32×C ÷ 32 = 32 ÷ 32 which means C = 1 making A = E = 7×C = 7 × 1 = 7 B = 9×C = 9 × 1 = 9 D = F = 4×C = 4 × 1 = 4 and ABCDEF = 791474