Puzzle for April 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.4, add D to both sides, and subtract C from each side: A – D + D – C = C + D – C which becomes eq.4a) A – C = D In eq.5, subtract C and D from each side of eq.5: C + F – A – C – D = A + D – C – D which becomes eq.5a) F – A – D = A – C
Hint #2
In eq.4a, replace A – C with F – A – D (from eq.5a): F – A – D = D Add A and D to both sides of the equation above: F – A – D + A + D = D + A + D which becomes eq.4b) F = A + 2×D
Hint #3
In eq.6, replace F with A + 2×D (from eq.4b): D + A + 2×D = A + B which becomes A + 3×D = A + B Subtract A from each side of the equation above: A + 3×D – A = A + B – A which makes 3×D = B
Hint #4
Subtract D from both sides of eq.6: D + F – D = A + B – D which becomes eq.6a) F = A + B – D In eq.3, substitute (A + B – D) for F (from eq.6a): B + D = A + C – (A + B – D) which is equivalent to B + D = A + C – A – B + D which becomes eq.3a) B + D = C – B + D
Hint #5
In eq.3a, subtract D from each side, and add B to each side: B + D – D + B = C – B + D – D + B which simplifies to 2×B = C Substitute (3×D) for B in the equation above: 2×(3×D) = C which makes 6×D = C
Hint #6
Substitute 6×D for C in eq.4: A – D = 6×D Add D to both sides of the equation above: A – D + D = 6×D + D which makes A = 7×D
Hint #7
Substitute 7×D for A in eq.4b: F = 7×D + 2×D which makes F = 9×D
Hint #8
Substitute 9×D for F, 3×D for B, and 6×D for C in eq.2: E + 9×D = 3×D + 6×D which becomes E + 9×D = 9×D Subtract 9×D from each side of the equation above: E + 9×D – 9×D = 9×D – 9×D which makes E = 0
Solution
Substitute 7×D for A, 3×D for B, 6×D for C, 0 for E, and 9×D for F into eq.1: 7×D + 3×D + 6×D + D + 0 + 9×D = 26 which simplifies to 26×D = 26 Divide both sides of the above equation by 26: 26×D ÷ 26 = 26 ÷ 26 which means D = 1 making A = 7×D = 7 × 1 = 7 B = 3×D = 3 × 1 = 3 C = 6×D = 6 × 1 = 6 F = 9×D = 9 × 1 = 9 and ABCDEF = 736109