Puzzle for March 20, 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 B with C + D (from eq.5): C + E = A + C + D Subtract C from each side of the equation above: C + E – C = A + C + D – C which becomes eq.2a) E = A + D
Hint #2
In eq.4, replace E with A + D (from eq.2a): D + F = A + D Subtract D from both sides of the above equation: D + F – D = A + D – D which makes F = A
Hint #3
Add F to both sides of eq.6: A + C + F + F = D – F + F which becomes A + C + 2×F = D In the equation above, substitute A for F: A + C + 2×A = D which becomes eq.6a) 3×A + C = D
Hint #4
Substitute 3×A + C for D (from eq.6a) in eq.2a: E = A + 3×A + C which becomes eq.2b) E = 4×A + C
Hint #5
Substitute 4×A + C for E (from eq.2b), and A for F in eq.3: 4×A + C + A = B – A which becomes 5×A + C = B – A Add A to both sides of the equation above: 5×A + C + A = B – A + A which becomes eq.3a) 6×A + C = B
Hint #6
Substitute 6×A + C for B (from eq.3a), and 3×A + C for D (from eq.6a) in eq.5: 6×A + C = C + 3×A + C which becomes 6×A + C = 2×C + 3×A Subtract C and 3×A from both sides of the equation above: 6×A + C – C – 3×A = 2×C + 3×A – C – 3×A which makes 3×A = C
Hint #7
Substitute 3×A for C in eq.3a: 6×A + 3×A = B which makes 9×A = B
Hint #8
Substitute 3×A for C in eq.6a: 3×A + 3×A = D which means 6×A = D
Hint #9
Substitute 3×A for C in eq.2b: E = 4×A + 3×A which makes E = 7×A
Solution
Substitute 9×A for B, 3×A for C, 6×A for D, and 7×A for E, and A for F in eq.1: A + 9×A + 3×A + 6×A + 7×A + A = 27 which simplifies to 27×A = 27 Divide both sides of the equation above by 27: 27×A ÷ 27 = 27 ÷ 27 which means A = 1 making B = 9×A = 9 × 1 = 9 C = 3×A = 3 × 1 = 3 D = 6×A = 6 × 1 = 6 E = 7×A = 7 × 1 = 7 F = A = 1 and ABCDEF = 193671