Puzzle for October 19, 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.3, replace C + F with A (from eq.2): E = A + A which makes eq.3a) E = 2×A
Hint #2
In eq.4, replace A with C + F (from eq.2): C - F = C + F - B + F which becomes C - F = C + 2×F - B In the equation above, subtract C from both sides, and add F and B to both sides: C - F - C + F + B = C + 2×F - B - C + F + B which simplifies to eq.4a) B = 3×F
Hint #3
In eq.5, substitute 2×A for E (from eq.3a), and 3×F for B (from eq.4a): D + 2×A = A + 3×F + F which becomes D + 2×A = A + 4×F Subtract A from both sides of the above equation: D + 2×A - A = A + 4×F - A which becomes eq.5a) D + A = 4×F
Hint #4
In eq.6, substitute 3×F for B (from eq.4a), and 2×A for E (from eq.3a): 3×F + D - A = A + C + 2×A which becomes 3×F + D - A = 3×A + C Add 2×A to both sides of the above equation: 3×F + D - A + 2×A = 3×A + C + 2×A which becomes eq.6a) 3×F + D + A = 5×A + C
Hint #5
Substitute 4×F for D + A (from eq.5a), and (C + F) for A (from eq.2) in eq.6a: 3×F + 4×F = 5×(C + F) + C which becomes 7×F = 5×C + 5×F + C which becomes 7×F = 6×C + 5×F Subtract 5×F from each side of the equation above: 7×F - 5×F = 6×C + 5×F - 5×F which makes 2×F = 6×C Divide both sides by 2: 2×F ÷ 2 = 6×C ÷ 2 which makes F = 3×C
Hint #6
Substitute 3×C for F in eq.2: A = C + 3×C which makes A = 4×C
Hint #7
Substitute (3×C) for F in eq.4a: B = 3×(3×C) which makes B = 9×C
Hint #8
Substitute (4×C) for A in eq.3a: E = 2×(4×C) which makes E = 8×C
Hint #9
Substitute 4×C for A, and (3×C) for F in eq.5a: D + 4×C = 4×(3×C) which becomes D + 4×C = 12×C Subtract 4×C from each side of the above equation: D + 4×C - 4×C = 12×C - 4×C which makes D = 8×C
Solution
Substitute 4×C for A, 9×C for B, 8×C for D and E, and 3×C for F in eq.1: 4×C + 9×C + C + 8×C + 8×C + 3×C = 33 which simplifies to 33×C = 33 Divide both sides of the above equation by 33: 33×C ÷ 33 = 33 ÷ 33 which means C = 1 making A = 4×C = 4 × 1 = 4 B = 9×C = 9 × 1 = 9 D = E = 8×C = 8 × 1 = 8 F = 3×C = 3 × 1 = 3 and ABCDEF = 491883