Puzzle for September 12, 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
eq.5 may be written as: B + F = A + D + C In the above equation, replace A + D with B + C (from eq.3): B + F = B + C + C which becomes B + F = B + 2×C Subtract B from each side: B + F - B = B + 2×C - B which makes eq.5a) F = 2×C
Hint #2
In eq.6, replace F with 2×C, and E with A + B (from eq.2): B + 2×C - A = A + C + A + B which becomes B + 2×C - A = 2×A + C + B Subtract B and C from both sides of the above equation: B + 2×C - A - B - C = 2×A + C + B - B - C which becomes C - A = 2×A Add A to both sides: C - A + A = 2×A + A which makes C = 3×A
Hint #3
In eq.5a, substitute (3×A) for C: F = 2×(3×A) which makes F = 6×A
Hint #4
Substitute 3×A for C, and 6×A for F in eq.4: 3×A + 6×A = A + E which becomes 9×A = A + E Subtract A from each side of the above equation: 9×A - A = A + E - A which makes 8×A = E
Hint #5
Substitute 8×A for E in eq.2: 8×A = A + B Subtract A from both sides of the above equation: 8×A - A = A + B - A which makes 7×A = B
Hint #6
Substitute 7×A for B, and 3×A for C in eq.3: 7×A + 3×A = A + D which becomes 10×A = A + D Subtract A from both sides of the equation above: 10×A - A = A + D - A which makes 9×A = D
Solution
Substitute 7×A for B, 3×A for C, 9×A for D, 8×A for E, and 6×A for F in eq.1: A + 7×A + 3×A + 9×A + 8×A + 6×A = 34 which simplifies to 34×A = 34 Divide both sides of the above equation by 34: 34×A ÷ 34 = 34 ÷ 34 which means A = 1 making B = 7×A = 7 × 1 = 7 C = 3×A = 3 × 1 = 3 D = 9×A = 9 × 1 = 9 E = 8×A = 8 × 1 = 8 F = 6×A = 6 × 1 = 6 and ABCDEF = 173986