Puzzle for September 23, 2020 ( )
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.3, replace B with A + C (from eq.6): A + C + C = A + E which becomes A + 2×C = A + E Subtract A from each side of the above equation: A + 2×C – A = A + E – A which makes eq.3a) 2×C = E
Hint #2
In eq.6, replace A with C + F (from eq.4): B = C + F + C which becomes eq.3b) B = 2×C + F
Hint #3
In eq.2, substitute 2×C + F for B (from eq.3b): C + D = 2×C + F + F which becomes C + D = 2×C + 2×F Subtract C from each side of the above equation: C + D – C = 2×C + 2×F – C which becomes eq.2a) D = C + 2×F
Hint #4
Substitute C + 2×F for D (from eq.2a) in eq.5: C + 2×F + F = C + E which becomes C + 3×F = C + E Subtract C from both sides of the above equation: C + 3×F – C = C + E – C which makes 3×F = E
Hint #5
Substitute 3×F for E in eq.3a: 2×C = 3×F Divide both sides of the above equation by 2: 2×C ÷ 2 = 3×F ÷ 2 which makes C = 1½×F
Hint #6
Substitute 1½×F for C in eq.4: A = 1½×F + F which makes A = 2½×F
Hint #7
Substitute 1½×F for C in eq.2a: D = 1½×F + 2×F which makes D = 3½×F
Hint #8
Substitute (1½×F) for C in eq.3b: B = 2×(1½×F) + F which is equivalent to B = 3×F + F which makes B = 4×F
Solution
Substitute 2½×F for A, 4×F for B, 1½×F for C, 3½×F for D, and 3×F for E in eq.1: 2½×F + 4×F + 1½×F + 3½×F + 3×F + F = 31 which simplifies to 15½×F = 31 Divide both sides of the above equation by 15½: 15½×F ÷ 15½ = 31 ÷ 15½ which means F = 2 making A = 2½×F = 2½ × 2 = 5 B = 4×F = 4 × 2 = 8 C = 1½×F = 1½ × 2 = 3 D = 3½×F = 3½ × 2 = 7 E = 3×F = 3 × 2 = 6 and ABCDEF = 583762