Puzzle for October 18, 2019 ( )
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 E with B + C (from eq.5): D + B + C = A In eq.6, replace the A at the right of the "=" sign with D + B + C: F – A = D + B + C – B – D which becomes F – A = C Add A to each side of the above equation: F – A + A = C + A which becomes eq.6a) F = C + A
Hint #2
In eq.3, substitute C + A for F (from eq.6a): C + C + A = A + D which becomes 2×C + A = A + D Subtract A from both sides of the above equation: 2×C + A – A = A + D – A which makes 2×C = D
Hint #3
Substitute A + D for C + F (from eq.3) in eq.2: B + E = A + D Subtract the left and right sides of eq.4 from the left and right sides of the equation above, respectively: B + E – (D + E) = A + D – A which becomes B + E – D – E = D which means B – D = D Add D to both sides of the above equation: B – D + D = D + D which means B = 2×D Substitute (2×C) for D: B = 2×(2×C) which makes B = 4×C
Hint #4
Substitute 4×C for B in eq.5: E = 4×C + C which makes E = 5×C
Hint #5
Substitute 2×C for D, and 5×C for E in eq.4: 2×C + 5×C = A which makes 7×C = A
Hint #6
Substitute 7×C for A in eq.6a: F = C + 7×C which makes F = 8×C
Solution
Substitute 7×C for A, 4×C for B, 2×C for D, 5×C for E, and 8×C for F in eq.1: 7×C + 4×C + C + 2×C + 5×C + 8×C = 27 which simplifies to 27×C = 27 Divide both sides of the equation above by 27: 27×C ÷ 27 = 27 ÷ 27 which means C = 1 making A = 7×C = 7 × 1 = 7 B = 4×C = 4 × 1 = 4 D = 2×C = 2 × 1 = 2 E = 5×C = 5 × 1 = 5 F = 8×C = 8 × 1 = 8 and ABCDEF = 741258