Puzzle for August 9, 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.6 may be written as: C + E + F = B + D - C In the above equation, replace B + D with A + E + F (from eq.5): C + E + F = A + E + F - C In the equation above, subtract E and F from both sides, and add C to both sides: C + E + F - E - F + C = A + E + F - C - E - F + C which simplifies to 2×C = A
Hint #2
In eq.2, replace A with 2×C: B - 2×C = 2×C - C which becomes B - 2×C = C Add 2×C to both sides of the above equation: B - 2×C + 2×C = C + 2×C which makes B = 3×C
Hint #3
In eq.3, substitute 3×C for B: E + F = 3×C + C which becomes eq.3a) E + F = 4×C
Hint #4
Substitute 4×C for E + F (from eq.3a), and 3×C for B in eq.6: C + 4×C = 3×C - C + D which becomes 5×C = 2×C + D Subtract 2×C from each side of the equation above: 5×C - 2×C = 2×C + D - 2×C which makes 3×C = D
Hint #5
Subtract F from both sides of eq.3a: E + F - F = 4×C - F which becomes eq.3b) E = 4×C - F
Hint #6
Substitute 3×C for D, 2×C for A, and 4×C - F for E (from eq.3b) in eq.4: 3×C + F = 2×C + 4×C - F which becomes 3×C + F = 6×C - F In the equation above, subtract 3×C from both sides, and add F to both sides: 3×C + F - 3×C + F = 6×C - F - 3×C + F which becomes 2×F = 3×C Divide both sides by 2: 2×F ÷ 2 = 3×C ÷ 2 which makes F = 1½×C
Hint #7
Substitute 1½×C for F in eq.3b: E = 4×C - 1½×C which makes E = 2½×C
Solution
Substitute 2×C for A, 3×C for B and D, 2½×C for E, and 1½×C for F in eq.1: 2×C + 3×C + C + 3×C + 2½×C + 1½×C = 26 which simplifies to 13×C = 26 Divide both sides of the above equation by 13: 13×C ÷ 13 = 26 ÷ 13 which means C = 2 making A = 2×C = 2 × 2 = 4 B = D = 3×C = 3 × 2 = 6 E = 2½×C = 2½ × 2 = 5 F = 1½×C = 1½ × 2 = 3 and ABCDEF = 462653