Puzzle for March 4, 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.4, replace B with D + F (from eq.3): D + F + C = A + F Subtract F from both sides of the above equation: D + F + C - F = A + F - F which becomes eq.4a) D + C = A
Hint #2
In eq.4a, replace A with D + E (from eq.2): D + C = D + E Subtract D from each side of the above equation: D + C - D = D + E - D which makes C = E
Hint #3
In eq.5, substitute C for E: D + C = C + F Subtract C from each side of the equation above: D + C - C = C + F - C which makes D = F
Hint #4
In eq.3, substitute F for D: F + F = B which makes 2×F = B
Hint #5
Substitute 2×F for B, and F for D in eq.6: A - 2×F = F ÷ F which means A - 2×F = 1 Add 2×F to each side of the equation above: A - 2×F + 2×F = 1 + 2×F which makes eq.6a) A = 1 + 2×F
Hint #6
Substitute F for D, and 1 + 2×F for A (from eq.6a) in eq.4a: F + C = 1 + 2×F Subtract F from each side: F + C - F = 1 + 2×F - F which means C = 1 + F which also means eq.4b) E = C = 1 + F
Solution
Substitute 1 + 2×F for A (from eq.6a), 2×F for B, 1 + F for C and E (from eq.4b), and F for D in eq.1: 1 + 2×F + 2×F + 1 + F + F + 1 + F + F = 27 which simplifies to 3 + 8×F = 27 Subtract 3 from each side of the above equation: 3 + 8×F - 3 = 27 - 3 which means 8×F = 24 Divide both sides by 8: 8×F ÷ 8 = 24 ÷ 8 which makes F = 3 making A = 1 + 2×F = 1 + 2×3 = 1 + 6 = 7 (from eq.6a) B = 2×F = 2×3 = 6 C = E = 1 + F = 1 + 3 = 4 (from eq.4b) D = F = 3 and ABCDEF = 764343