Puzzle for October 27, 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
Help Area
Hint #1
eq.6 may be written as: A + C + D = B + F – D + E In the above equation, replace B + F with A + D (from eq.3): A + C + D = A + D – D + E which becomes A + C + D = A + E Subtract A from both sides of the above equation: A + C + D – A = A + E – A which becomes eq.6a) C + D = E
Hint #2
In eq.2, replace E with C + D (from eq.6a): C + C + D = B + D which becomes 2×C + D = B + D Subtract D from each side of the equation above: 2×C + D – D = B + D – D which makes 2×C = B
Hint #3
In eq.5, substitute 2×C for B: 2×C + E = C + F Subtract C from both sides of the equation above: 2×C + E – C = C + F – C which becomes eq.5a) C + E = F
Hint #4
In eq.2, substitute F for C + E (from eq.5a): eq.2a) F = B + D
Hint #5
Substitute B + D for F (from eq.2a) in eq.3: A + D = B + B + D which becomes A + D = 2×B + D Subutract D from both sides of the above equation: A + D – D = 2×B + D – D which makes eq.3a) A = 2×B
Hint #6
Substitute 2×C for B in eq.3a: A = 2×(2×C) which makes A = 4×C
Hint #7
Substitute C + D for E (from eq.6a), 4×C for A, and 2×C for B in eq.4: D + C + D = 4×C + 2×C which becomes 2×D + C = 6×C Subtract C from each side of the above equation: 2×D + C – C = 6×C – C which makes 2×D = 5×C Divide both sides by 2: 2×D ÷ 2 = 5×C ÷ 2 which makes D = 2½×C
Hint #8
Substitute 2½×C for D in eq.6a: C + 2½×C = E which makes 3½×C = E
Hint #9
Substitute 3½×C for E in eq.5a: C + 3½×C = F which makes 4½×C = F
Solution
Substitute 4×C for A, 2×C for B, 2½×C for D, 3½×C for E, and 4½×C for F in eq.1: 4×C + 2×C + C + 2½×C + 3½×C + 4½×C = 35 which simplifies to 17½×C = 35 Divide both sides of the above equation by 17½: 17½×C ÷ 17½ = 35 ÷ 17½ which means C = 2 making A = 4×C = 4 × 2 = 8 B = 2×C = 2 × 2 = 4 D = 2½×C = 2½ × 2 = 5 E = 3½×C = 3½ × 2 = 7 F = 4½×C = 4½ × 2 = 9 and ABCDEF = 842579