Puzzle for January 23, 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 positive integer.
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Hint #1
In eq.4, replace A with B + C (from eq.1): C - B = B + C - C which becomes C - B = B Add B to both sides of the above equation: C - B + B = B + B which makes C = 2×B
Hint #2
In eq.1, replace C with 2×B: A = B + 2×B which makes A = 3×B
Hint #3
In eq.2, substitute 3×B for A: F = 3×B + B which makes F = 4×B
Hint #4
Substitute 4×B for F, 3×B for A, and 2×B for C in eq.6: B × 4×B = 3×B - B + 2×B which becomes 4×B² = 4×B Divide both sides of the above equation by 4×B: 4×B² ÷ 4×B = 4×B ÷ 4×B which makes B = 1 making A = 3×B = 3 × 1 = 3 C = 2×B = 2 × 1 = 2 F = 4×B = 4 × 1 = 4
Hint #5
Substitute 4 for F, and 2 for C in eq.3: E + 4 = 2 + D Subtract 2 from each side of the above equation: E + 4 - 2 = 2 + D - 2 which makes eq.3a) E + 2 = D
Hint #6
Substitute 2 for C, 1 for B, and E + 2 for D (from eq.3a) in eq.5: 2 × E = 1 + E + 2 which becomes 2×E = 3 + E Subtract E from each side of the equation above: 2×E - E = 3 + E - E which makes E = 3
Solution
Substitute 3 for E in eq.3a: 3 + 2 = D which makes 5 = D and makes ABCDEF = 312534