Puzzle for September 20, 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
In eq.5, replace B + D with A + F (from eq.3): E + F - D = A + F In the above equation, subtract F from both sides, and add D to both sides: E + F - D - F + D = A + F - F + D which becomes eq.5a) E = A + D
Hint #2
In eq.2, replace E with A + D (from eq.5a): A + D = C + D Subtract D from each side of the equation above: A + D - D = C + D - D which makes A = C
Hint #3
Subtract A from each side of eq.5a: E - A = A + D - A which becomes eq.5b) E - A = D
Hint #4
In eq.4, substitute E - A for D (from eq.5b): E - A - E = E - F which becomes -A = E - F Add A and F to both sides of the above equation: -A + A + F = E - F + A + F which becomes eq.4a) F = E + A
Hint #5
Substitute (E - A) for D (from eq.5b), and E + A for F (from eq.4a) in eq.5: B + (E - A) = E + E + A - (E - A) which becomes B + E - A = 2×E + A - E + A which becomes B + E - A = E + 2×A In the above equation, subtract E from both sides, and add A to both sides: B + E - A - E + A = E + 2×A - E + A which makes B = 3×A
Hint #6
Substitute 3×A for B, E + A for F (from eq.4a), and E - A for D (from eq.5b) in eq.6: 3×A + E + A = E - A + E - 3×A which becomes 4×A + E = 2×E - 4×A In the above equation, subtract E from both sides, and add 4×A to both sides: 4×A + E - E + 4×A = 2×E - 4×A - E + 4×A which simplifies to 8×A = E
Hint #7
Substitute 8×A for E in eq.5b: 8×A - A = D which makes 7×A = D
Hint #8
Substitute 8×A for E in eq.4a: F = 8×A + A which makes F = 9×A
Solution
Substitute 3×A for B, A for C, 7×A for D, 8×A for E, and 9×A for F in eq.1: A + 3×A + A + 7×A + 8×A + 9×A = 29 which simplifies to 29×A = 29 Divide both sides of the above equation by 29: 29×A ÷ 29 = 29 ÷ 29 which means A = 1 making B = 3×A = 3 × 1 = 3 C = A = 1 D = 7×A = 7 × 1 = 7 E = 8×A = 8 × 1 = 8 F = 9×A = 9 × 1 = 9 and ABCDEF = 131789