Puzzle for June 22, 2020 ( )
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.2, replace A with D + F (from eq.4): B + D = D + F + F which becomes B + D = D + 2×F Subtract D from each side of the above equation: B + D – D = D + 2×F – D which makes eq.2a) B = 2×F
Hint #2
In eq.5, replace E + F with A + C (from eq.3): B + C = A + A + C which becomes B + C = 2×A + C Subtract C from both sides of the above equation: B + C – C = 2×A + C – C which makes eq.5a) B = 2×A
Hint #3
In eq.2a, substitute 2×A for B (from eq.5a): 2×A = 2×F Divide both sides of the equation above by 2: 2×A ÷ 2 = 2×F ÷ 2 which makes A = F
Hint #4
Substitute A for F in eq.4: D + A = A Subtract A from both sides of the equation above: D + A – A = A – A which makes D = 0
Hint #5
Substitute A for F in eq.3: E + A = A + C Subtract A from each side of the equation above: E + A – A = A + C – A which makes E = C
Hint #6
Substitute A for F, and 2×A for B in eq.6: C + A = 2×A – C In the equation above, subtract A from both sides, and add C to both sides: C + A – A + C = 2×A – C – A + C which makes 2×C = A and also makes F = A = 2×C
Hint #7
Substitute (2×C) for A in eq.5a: B = 2×(2×C) which makes B = 4×C
Solution
Substitute 2×C for A and F, 4×C for B, 0 for D, and C for E in eq.1: 2×C + 4×C + C + 0 + C + 2×C = 10 which simplifies to 10×C = 10 Divide both sides of the equation above by 10: 10×C ÷ 10 = 10 ÷ 10 which means C = 1 making A = F = 2×C = 2 × 1 = 2 B = 4×C = 4 × 1 = 4 E = C = 1 and ABCDEF = 241012