Puzzle for March 22, 2021 ( )
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.4, replace B with D + E (from eq.3): D + E + D = A + E which becomes 2×D + E = A + E Subtract E from each side of the equation above: 2×D + E – E = A + E – E which makes eq.4a) 2×D = A
Hint #2
In eq.5, replace A with 2×D: C + D = 2×D – C In the equation above, subtract D from both sides, and add C to both sides: C + D – D + C = 2×D – C – D + C which makes 2×C = D
Hint #3
In eq.4a, substitute (2×C) for D: 2×(2×C) = A which makes 4×C = A
Hint #4
Substitute 4×C for A in eq.6: 4×C – E = C + E In the above equation, add E to both sides, and subtract C from both sides: 4×C – E + E – C = C + E + E – C which makes 3×C = 2×E Divide both sides by 2: 3×C ÷ 2 = 2×E ÷ 2 which makes 1½×C = E
Hint #5
Substitute 2×C for D, and 1½×C for E in eq.3: 2×C + 1½×C = B which makes 3½×C = B
Hint #6
Substitute 3½×C for B in eq.2: F = 3½×C + C which makes F = 4½×C
Solution
Substitute 4×C for A, 3½×C for B, 2×C for D, 1½×C for E, and 4½×C for F in eq.1: 4×C + 3½×C + C + 2×C + 1½×C + 4½×C = 33 which simplifies to 16½×C = 33 Divide both sides of the equation above by 16½: 16½×C ÷ 16½ = 33 ÷ 16½ which means C = 2 making A = 4×C = 4 × 2 = 8 B = 3½×C = 3½ × 2 = 7 D = 2×C = 2 × 2 = 4 E = 1½×C = 1½ × 2 = 3 F = 4½×C = 4½ × 2 = 9 and ABCDEF = 872439