Puzzle for January 26, 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.6, replace C + F with A + D (from eq.3): D + E = A + A + D which becomes D + E = 2×A + D Subtract D from each side of the equation above: D + E – D = 2×A + D – D which makes E = 2×A
Hint #2
In eq.4, replace E with 2×A: A + 2×A = D – A which becomes 3×A = D – A Add A to both sides of the equation above: 3×A + A = D – A + A which makes 4×A = D
Hint #3
In eq.3, substitute A + B for C (from eq.5): A + B + F = A + D Subtract A from each side of the above equation: A + B + F – A = A + D – A which becomes eq.3a) B + F = D
Hint #4
Substitute B + F for D (from eq.3a) in eq.2: B + C = B + F Subtract B from each side of the equation above: B + C – B = B + F – B which makes C = F
Hint #5
Substitute C for F, and 4×A for D in eq.3: C + C = A + 4×A which becomes 2×C = 5×A Divide both sides of the above equation by 2: 2×C ÷ 2 = 5×A ÷ 2 which makes C = 2½×A and also makes F = C = 2½×A
Hint #6
Substitute 2½×A for C in eq.5: 2½×A = A + B Subtract A from each side of the equation above: 2½×A – A = A + B – A which makes 1½×A = B
Solution
Substitute 1½×A for B, 2½×A for C and F, 4×A for D, and 2×A for E in eq.1: A + 1½×A + 2½×A + 4×A + 2×A + 2½×A = 27 which simplifies to 13½×A = 27 Divide both sides of the above equation by 13½: 13½×A ÷ 13½ = 27 ÷ 13½ which means A = 2 making B = 1½×A = 1½ × 2 = 3 C = F = 2½×A = 2½ × 2 = 5 D = 4×A = 2 × 2 = 8 E = 2×A = 2 × 2 = 4 and ABCDEF = 235845