Puzzle for September 1, 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.5, replace B with C + F (from eq.2): A + F = C + F + C which becomes A + F = 2×C + F Subtract F from both sides of the above equation: A + F – F = 2×C + F – F which makes A = 2×C
Hint #2
Add D to both sides of eq.6: B – D + D = A + D + D which becomes eq.6a) B = A + 2×D In eq.4, replace B with A + 2×D (from eq.6a): E + F = A + 2×D + D which becomes eq.4a) E + F = A + 3×D
Hint #3
In eq.4a, substitute A + E for F (from eq.3): E + A + E = A + 3×D which becomes 2×E + A = A + 3×D Subtract A from each side of the above equation: 2×E + A – A = A + 3×D – A which becomes 2×E = 3×D Divide both sides by 2: 2×E ÷ 2 = 3×D ÷ 2 which makes eq.4b) E = 1½×D
Hint #4
In eq.4, substitute 1½×D for E, and C + F for B (from eq.2): 1½×D + F = C + F + D Subtract F and D from both sides of the equation above: 1½×D + F – F – D = C + F + D – F – D which makes ½×D = C Multiply both sides by 2: ½×D × 2 = C × 2 which makes D = 2×C
Hint #5
Substitute (2×C) for D in eq.4b: E = 1½×(2×C) which makes E = 3×C
Hint #6
Substitute (2×C) for A and D in eq.6a: B = (2×C) + 2×(2×C) which becomes B = 2×C + 4×C which makes B = 6×C
Hint #7
Substitute 2×C for A, and 3×C for E in eq.3: F = 2×C + 3×C which makes F = 5×C
Solution
Substitute 2×C for A and D, 6×C for B, 3×C for E, and 5×C for F in eq.1: 2×C + 6×C + C + 2×C + 3×C + 5×C = 19 which simplifies to 19×C = 19 Divide both sides of the above equation by 19: 19×C ÷ 19 = 19 ÷ 19 which means C = 1 making A = D = 2×C = 2 × 1 = 2 B = 6×C = 6 × 1 = 6 E = 3×C = 3 × 1 = 3 F = 5×C = 5 × 1 = 5 and ABCDEF = 261235