Puzzle for May 12, 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
Help Area
Hint #1
Subtract the left and right sides of eq.3 from the left and right sides of eq.2, respectively: B + E - (D + E) = A + F - (A + C) which becomes B + E - D - E = A + F - A - C which becomes B - D = F - C Add D and C to both sides of the above equation: B - D + D + C = F - C + D + C which becomes eq.2a) B + C = F + D
Hint #2
Add F and A to both sides of eq.4: A - F + F + A = D - A + F + A which becomes 2×A = D + F which may be written as eq.4a) 2×A = F + D
Hint #3
In eq.4a, replace F + D with B + C (from eq.2a): eq.4b) 2×A = B + C
Hint #4
eq.5 may be written as: E = (B + C + D + F) ÷ 4 Multiply both sides of the above equation by 4: 4 × E = 4 × (B + C + D + F) ÷ 4 which becomes 4×E = B + C + D + F which may be written as eq.5a) 4×E = B + C + F + D
Hint #5
In eq.5a, replace B + C with 2×A (from eq.4b), and F + D with 2×A (from eq.4a): 4×E = 2×A + 2×A which becomes 4×E = 4×A Divide both sides of the above equation by 4: 4×E ÷ 4 = 4×A ÷ 4 which makes E = A
Hint #6
In eq.3, substitute A for E: D + A = A + C Subtract A from each side of the above equation: D + A - A = A + C - A which makes D = C
Hint #7
Substitute C for D in eq.2a: B + C = F + C Subtract C from each side of the equation above: B + C - C = F + C - C which makes B = F
Hint #8
Substitute C for D, and A for E in eq.6: C + C = B + (A ÷ A) which becomes 2×C = B + 1 Subtract 1 from each side of the above equation: 2×C - 1 = B + 1 - 1 which makes 2×C - 1 = B and also makes eq.6a) 2×C - 1 = B = F
Hint #9
Substitute 2×C - 1 for B (from eq.6a) into eq.4b: 2×A = 2×C - 1 + C which becomes 2×A = 3×C - 1 Divide both sides of the above equation by 2: (2×A) ÷ 2 = (3×C - 1) ÷ 2 which makes A = 1½×C - ½ and also makes eq.2b) E = A = 1½×C - ½
Hint #10
Substitute 1½×C - ½ for A and E (from eq.2b), 2×C - 1 for B and F (from eq.6a), and C for D in eq.1: 1½×C - ½ + 2×C - 1 + C + C + 1½×C - ½ + 2×C - 1 = 42 which simplifies to 9×C - 3 = 42 Add 3 to both sides of the above equation: 9×C - 3 + 3 = 42 + 3 which makes 9×C = 45 Divide both sides by 9: 9×C ÷ 9 = 45 ÷ 9 which makes C = 5
Solution
Since C = 5, then: A = E = 1½×C - ½ = 1½×5 - ½ = 7½ - ½ = 7 (from eq.2b) B = F = 2×C - 1 = 2×5 - 1 = 10 - 1 = 9 (from eq.6a) D = C = 5 and ABCDEF = 795579