Puzzle for October 5, 2022 ( )
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.4, replace F with A + C (from eq.2): C + A + C = A + D which becomes 2×C + A = A + D Subtract A from each side of the above equation: 2×C + A - A = A + D - A which makes 2×C = D
Hint #2
In eq.3, replace D with 2×C: E = C + 2×C which makes E = 3×C
Hint #3
In eq.6, replace D with 2×C: F - 2×C = B - F Add F to both sides of the above equation: F - 2×C + F = B - F + F which becomes eq.6a) 2×F - 2×C = B
Hint #4
In eq.5, substitute 3×C for E: B - F = 3×C - B Add F and B to both sides of the above equation: B - F + F + B = 3×C - B + F + B which becomes eq.5a) 2×B = 3×C + F
Hint #5
Substitute (2×F - 2×C) for B (from eq.6a) into eq.5a: 2×(2×F - 2×C) = 3×C + F which becomes 4×F - 4×C = 3×C + F In the above equation, add 4×C to both sides, and subtract F from both sides: 4×F - 4×C + 4×C - F = 3×C + F + 4×C - F which makes 3×F = 7×C Divide both sides by 3: 3×F ÷ 3 = 7×C ÷ 3 which makes F = 2⅓×C
Hint #6
Substitute 2⅓×C for F in eq.2: 2⅓×C = A + C Subtract C from each side of the equation above: 2⅓×C - C = A + C - C which makes 1⅓×C = A
Hint #7
Substitute (2⅓×C) for F in eq.6a: 2×(2⅓×C) - 2×C = B which becomes 4⅔×C - 2×C = B which makes 2⅔×C = B
Solution
Substitute 1⅓×C for A, 2⅔×C for B, 2×C for D, 3×C for E, and 2⅓×C for F in eq.1: 1⅓×C + 2⅔×C + C + 2×C + 3×C + 2⅓×C = 37 which simplifies to 12⅓×C = 37 Divide both sides of the above equation by 12⅓: 12⅓×C ÷ 12⅓ = 37 ÷ 12⅓ which means C = 3 making A = 1⅓×C = 1⅓ × 3 = 4 B = 2⅔×C = 2⅔ × 3 = 8 D = 2×C = 2 × 3 = 6 E = 3×C = 3 × 3 = 9 F = 2⅓×C = 2⅓ × 3 = 7 and ABCDEF = 483697