Puzzle for March 27, 2019 ( )
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 F with A + E (from eq.5): A + A + E = D + E Subtract E from both sides of the equation above: A + A + E - E = D + E - E which makes 2×A = D
Hint #2
In eq.3, replace E with C + D (from eq.2): B + C + D = D + F Subtract D from each side of the above equation: B + C + D - D = D + F - D which becomes eq.2a) B + C = F
Hint #3
In eq.4, substitute B + C for F (from eq.2a): C + E = B + B + C Subtract C from each side of the above equation: C + E - C = B + B + C - C which makes eq.3a) E = 2×B
Hint #4
Substitute 2×A for D, and 2×B for E into eq.2: eq.2b) 2×B = C + 2×A
Hint #5
Substitute A + E for F (from eq.5) into eq.4: C + E = B + A + E Subtract E from each side of the above equation: C + E - E = B + A + E - E which becomes eq.4a) C = B + A
Hint #6
Substitute B + A for C (from eq.4a) in eq.2b: 2×B = B + A + 2×A Subtract B from both sides: 2×B - B = B + A + 2×A - B which makes B = 3×A
Hint #7
Substitute (3×A) for B in eq.3a: E = 2×(3×A) which means E = 6×A
Hint #8
Substitute 3×A for B in eq.4a: C = 3×A + A which makes C = 4×A
Hint #9
Substitute 6×A for E in eq.5: F = A + 6×A which makes F = 7×A
Solution
Substitute 3×A for B, 4×A for C, 2×A for D, 6×A for E, and 7×A for F in eq.1: A + 3×A + 4×A + 2×A + 6×A + 7×A = 23 which becomes 23×A = 23 Divide both sides by 23: 23×A ÷ 23 = 23 ÷ 23 which means A = 1 making B = 3×A = 3 × 1 = 3 C = 4×A = 4 × 1 = 4 D = 2×A = 2 × 1 = 2 E = 6×A = 6 × 1 = 6 F = 7×A = 7 × 1 = 7 and ABCDEF = 134267