Puzzle for March 15, 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
Add D and F to both sides of eq.4: E – D + D + F = D – F + D + F which becomes eq.4a) E + F = 2×D In eq.6, replace E + F with 2×D (from eq.4a): C + D = A + 2×D Subtract D from each side of the equation above: C + D – D = A + 2×D – D which becomes eq.6a) C = A + D
Hint #2
In eq.3, replace A + D with C (from eq.6a): B + C = C Subtract C from each side of the above equation: B + C – C = C – C which makes B = 0
Hint #3
In eq.5, substitute 0 for B: F – A = A – 0 which becomes F – A = A Add A to both sides of the above equation: F – A + A = A + A which makes F = 2×A
Hint #4
Substitute 2×A for F, and 0 for B in eq.2: D = A + 2×A – 0 which makes D = 3×A
Hint #5
Substitute 3×A for D in eq.6a: C = A + 3×A which makes C = 4×A
Hint #6
Substitute 2×A for F, and (3×A) for D in eq.4a: E + 2×A = 2×(3×A) which becomes E + 2×A = 6×A Subtract 2×A from both sides of the equation above: E + 2×A – 2×A = 6×A – 2×A which makes E = 4×A
Solution
Substitute 0 for B, 4×A for C and E, 3×A for D, and 2×A for F in eq.1: A + 0 + 4×A + 3×A + 4×A + 2×A = 14 which simplifies to 14×A = 14 Divide both sides of the above equation by 14: 14×A ÷ 14 = 14 ÷ 14 which means A = 1 making C = E = 4×A = 4 × 1 = 4 D = 3×A = 3 × 1 = 3 F = 2×A = 2 × 1 = 2 and ABCDEF = 104342