Puzzle for November 9, 2021 ( )
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.5 may be written as: C + E = A + F + B In the above equation, replace E with B + C (from eq.2), and A + F with D (from eq.3): C + B + C = D + B which becomes 2×C + B = D + B Subtract B from each side of the equation above: 2×C + B – B = D + B – B which becomes 2×C = D
Hint #2
eq.6 may be written as: B – D = A + F + C + D In the equation above, replace A + F with D (from eq.3): B – D = D + C + D which becomes B – D = 2×D + C Add D to both sides: B – D + D = 2×D + C + D which becomes eq.6a) B = 3×D + C
Hint #3
In eq.6a, substitute (2×C) for D: B = 3×(2×C) + C which becomes B = 6×C + C which makes B = 7×C
Hint #4
Substitute 7×C for B in eq.2: E = 7×C + C which makes E = 8×C
Hint #5
Substitute 7×C for B, 8×C for E, and 2×C for D in eq.4: 7×C – F = 8×C – 2×C which becomes 7×C – F = 6×C In the above equation, add F to both sides, and subtract 6×C from both sides: 7×C – F + F – 6×C = 6×C + F – 6×C which becomes C = F
Hint #6
Substitute 2×C for D, and C for F in eq.3: 2×C = A + C Subtract C from each side of the equation above: 2×C – C = A + C – C which makes C = A
Solution
Substitute C for A and F, 7×C for B, 2×C for D, and 8×C for E in eq.1: C + 7×C + C + 2×C + 8×C + C = 20 which simplifies to 20×C = 20 Divide both sides of the above equation by 20: 20×C ÷ 20 = 20 ÷ 20 which means C = 1 making A = F = C = 1 B = 7×C = 7 × 1 = 7 D = 2×C = 2 × 1 = 2 E = 8×C = 8 × 1 = 8 and ABCDEF = 171281