Puzzle for January 26, 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
In eq.4, replace A with C + E (from eq.2): B + C = C + E + E which becomes B + C = C + 2×E Subtract C from each side of the equation above: B + C - C = C + 2×E - C which makes B = 2×E
Hint #2
In eq.5, replace D with C + E + F (from eq.3), A with C + E (from eq.2), and B with 2×E: C + E + F + F = C + E + 2×E + C which becomes C + E + 2×F = 2×C + 3×E Subtract C and 3×E from each side of the above equation: C + E + 2×F - C - 3×E = 2×C + 3×E - C - 3×E which becomes eq.5a) 2×F - 2×E = C
Hint #3
In eq.2, substitute 2×F - 2×E for C (from eq.5a): A = 2×F - 2×E + E which becomes eq.2a) A = 2×F - E
Hint #4
In eq.3, substitute 2×F - 2×E for C (from eq.5a): D = 2×F - 2×E + E + F which becomes eq.3a) D = 3×F - E
Hint #5
eq.6 may be written as: B = (D + E + F) ÷ 3 Multiply both sides of the above equation by 3: 3 × B = 3 × (D + E + F) ÷ 3 which becomes eq.6a) 3×B = D + E + F
Hint #6
Substitute (2×E) for B, and 3×F - E for D (from eq.3a) into eq.6a: 3×(2×E) = 3×F - E + E + F which makes 6×E = 4×F Divide both sides of the above equation by 4: 6×E ÷ 4 = 4×F ÷ 4 which makes 1½×E = F
Hint #7
Substitute (1½×E) for F in eq.5a: 2×(1½×E) - 2×E = C which becomes 3×E - 2×E = C which makes E = C
Hint #8
Substitute E for C in eq.2: A = E + E which makes A = 2×E
Hint #9
Substitute (1½×E) for F in eq.3a: D = 3×(1½×E) - E which becomes D = 4½×E - E which makes D = 3½×E
Solution
Substitute 2×E for A and B, E for C, 3½×E for D, and 1½×E for F in eq.1: 2×E + 2×E + E + 3½×E + E + 1½×E = 22 which simplifies to 11×E = 22 Divide both sides of the above equation by 11: 11×E ÷ 11 = 22 ÷ 11 which means E = 2 making A = B = 2×E = 2 × 2 = 4 C = E = 2 D = 3½×E = 3½ × 2 = 7 F = 1½×E = 1½ × 2 = 3 and ABCDEF = 442723