Puzzle for July 30, 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
Subtract the left and right sides of eq.6 from the left and right sides of eq.5, respectively: E – A – (D – A) = A + D – F – (A + B – D – F) which becomes E – A – D + A = A + D – F – A – B + D + F which becomes E – D = 2×D – B Add D to both sides of the above equation: E – D + D = 2×D – B + D which becomes eq.5a) E = 3×D – B
Hint #2
In eq.4, substitute (3×D – B) for E (from eq.5a): D – (3×D – B) = B – C which becomes D – 3×D + B = B – C which becomes –2×D + B = B – C Subtract B from each side of the equation above: –2×D + B – B = B – C – B which becomes –2×D = –C Multiply both sides by (–1): –2×D × (–1) = –C × (–1) which makes 2×D = C
Hint #3
Subtract the left and right sides of eq.3 from the left and right sides of eq.2, respectively: B – E – (C – E) = A – D – (A – C) which becomes B – E – C + E = A – D – A + C which becomes B – C = –D + C Add C to both sides of the equation above: B – C + C = –D + C + C which becomes B = –D + 2×C which may be written as eq.2a) B = 2×C – D
Hint #4
In eq.2a, substitute (2×D) for C: B = 2×(2×D) – D which becomes B = 4×D – D which makes B = 3×D
Hint #5
Substitute 3×D for B in eq.5a: E = 3×D – 3×D which makes E = 0
Hint #6
Substitute 3×D for B, and 0 for E in eq.2: 3×D – 0 = A – D which becomes 3×D = A – D Add D to both sides of the above equation: 3×D + D = A – D + D which makes 4×D = A
Hint #7
Substitute 0 for E, and 4×D for A in eq.5: 0 – 4×D = 4×D + D – F which becomes –4×D = 5×D – F Add 4×D and F to both sides of the equation above: –4×D + 4×D + F = 5×D – F + 4×D + F which makes F = 9×D
Solution
Substitute 4×D for A, 3×D for B, 2×D for C, 0 for E, and 9×D for F in eq.1: 4×D + 3×D + 2×D + D + 0 + 9×D = 19 which simplifies to 19×D = 19 Divide both sides of the above equation by 19: 19×D ÷ 19 = 19 ÷ 19 which means D = 1 making A = 4×D = 4 × 1 = 4 B = 3×D = 3 × 1 = 3 C = 2×D = 2 × 1 = 2 F = 9×D = 9 × 1 = 9 and ABCDEF = 432109