Puzzle for November 2, 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
Subtract C and A from both sides of eq.3: C + E - C - A = A + D - C - A which becomes eq.3a) E - A = D - C Add D to both sides of eq.5: E - C + D = B - D + D which becomes E - C + D = B which may be written as eq.5a) D - C + E = B
Hint #2
In eq.5a, replace D - C with E - A (from eq.3a): E - A + E = B which becomes 2×E - A = B In the above equation, add A to both sides, and subtract B from both sides: 2×E - A + A - B = B + A - B which becomes eq.5b) 2×E - B = A
Hint #3
In eq.4, replace A with 2×E - B (from eq.5b): B - E = 2×E - B - B which becomes B - E = 2×E - 2×B Add E and 2×B to both sides of the above equation: B - E + E + 2×B = 2×E - 2×B + E + 2×B which becomes 3×B = 3×E Divide both sides by 3: 3×B ÷ 3 = 3×E ÷ 3 which makes B = E
Hint #4
In eq.5b, replace E with B: 2×B - B = A which makes B = A and also makes E = B = A
Hint #5
In eq.5, substitute B for E: B - C = B - D Subtract B from each side of the above equation: B - C - B = B - D - B which makes -C = -D which means C = D
Hint #6
Substitute C for D in eq.2: F = C + C which makes F = 2×C
Hint #7
Substitute 2×C for F, and A for B in eq.6: C + 2×C = A + A - 2×C which becomes 3×C = 2×A - 2×C Add 2×C to both sides of the equation above: 3×C + 2×C = 2×A - 2×C + 2×C which makes 5×C = 2×A Divide both sides by 2: 5×C ÷ 2 = 2×A ÷ 2 which makes 2½×C = A and also makes E = B = A = 2½×C
Solution
Substitute 2½×C for A and B and E, C for D, and 2×C for F in eq.1: 2½×C + 2½×C + C + C + 2½×C + 2×C = 23 which simplifies to 11½×C = 23 Divide both sides of the above equation by 11½: 11½×C ÷ 11½ = 23 ÷ 11½ which means C = 2 making A = B = E = 2½×C = 2½ × 2 = 5 D = C = 2 F = 2×C = 2 × 2 = 4 and ABCDEF = 552254