Puzzle for March 13, 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 C with E + F (from eq.4): E + F + E = D which becomes eq.2a) 2×E + F = D
Hint #2
In eq.5, replace D with 2×E + F (from eq.2a): 2×E + F = A + E + F Subtract E and F from both sides of the above equation: 2×E + F – E – F = A + E + F – E – F which simplifies to E = A
Hint #3
In eq.3, substitute A for E: F = A + B + A which becomes eq.3a) F = 2×A + B
Hint #4
Substitute A for E, and 2×A + B for F (from eq.3a) in eq.4: A + 2×A + B = C which becomes eq.4a) 3×A + B = C
Hint #5
Substitute A for E, and 2×A + B for F (from eq.3a) in eq.5: D = A + A + 2×A + B which becomes eq.5a) D = 4×A + B
Hint #6
Substitute 3×A + B for C (from eq.4a), 2×A + B for F (from eq.3a), 4×A + B for D (from eq.5a), and A for E in eq.6: B + 3×A + B + 2×A + B = A + 4×A + B + A which becomes 5×A + 3×B = 6×A + B Subtract 5×A and B from both sides of the above equation: 5×A + 3×B – 5×A – B = 6×A + B – 5×A – B which makes 2×B = A and also makes E = A = 2×B
Hint #7
Substitute (2×B) for A in eq.4a: 3×(2×B) + B = C which becomes 6×B + B = C which makes 7×B = C
Hint #8
Substitute (2×B) for A in eq.5a: D = 4×(2×B) + B which becomes D = 8×B + B which makes D = 9×B
Hint #9
Substitute (2×B) for A in eq.3a: F = 2×(2×B) + B which becomes F = 4×B + B which makes F = 5×B
Solution
Substitute 2×B for A and E, 7×B for C, 9×B for D, and 5×B for F in eq.1: 2×B + B + 7×B + 9×B + 2×B + 5×B = 26 which simplifies to 26×B = 26 Divide both sides of the equation above by 26: 26×B ÷ 26 = 26 ÷ 26 which means B = 1 making A = E = 2×B = 2 × 1 = 2 C = 7×B = 7 × 1 = 7 D = 9×B = 9 × 1 = 9 F = 5×B = 5 × 1 = 5 and ABCDEF = 217925