Puzzle for June 29, 2023 ( )
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
Subtract the left and right sides of eq.6 from the left and right sides of eq.5, respectively: B + D - F - (A + D - F) = C + F - (C - A) which becomes B + D - F - A - D + F = C + F - C + A which becomes B - A = F + A Add A to both sides of the above equation: B - A + A = F + A + A which becomes eq.5a) B = F + 2×A
Hint #2
In eq.2, replace B with F + 2×A (from eq.5a): F + 2×A = E + F Subtract F from both sides of the equation above: F + 2×A - F = E + F - F which makes 2×A = E
Hint #3
In eq.3, replace E with 2×A: C + 2×A = A + F Subtract A from each side of the equation above: C + 2×A - A = A + F - A which becomes eq.3a) C + A = F
Hint #4
In eq.4, substitute C + A for F (from eq.3a): D + C + A = A + B + C Subtract C and A from both sides of the above equation: D + C + A - C - A = A + B + C - C - A which simplifies to D = B
Hint #5
Substitute D for B in eq.5a: eq.5b) D = F + 2×A
Hint #6
Substitute F + 2×A for B (from eq.5a) and for D (from eq.5b) in eq.5: F + 2×A + F + 2×A - F = C + F which becomes F + 4×A = C + F Subtract F from each side of the above equation: F + 4×A - F = C + F - F which makes 4×A = C
Hint #7
Substitute 4×A for C in eq.3a: 4×A + A = F which makes 5×A = F
Hint #8
Substitute 5×A for F in eq.5a: B = 5×A + 2×A which makes B = 7×A and also makes D = B = 7×A
Solution
Substitute 7×A for B and D, 4×A for C, 2×A for E, and 5×A for F in eq.1: A + 7×A + 4×A + 7×A + 2×A + 5×A = 26 which simplifies to 26×A = 26 Divide both sides of the above equation by 26: 26×A ÷ 26 = 26 ÷ 26 which means A = 1 making B = D = 7×A = 7 × 1 = 7 C = 4×A = 4 × 1 = 4 E = 2×A = 2 × 1 = 2 F = 5×A = 5 × 1 = 5 and ABCDEF = 174725