Puzzle for January 22, 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
Subtract both B and D from each side of eq.3: B + F - B - D = A + D - B - D which reduces to F - D = A - B In the equation above, substitute (A + B) for D (from eq.2): F - (A + B) = A - B which is the same as F - A - B = A - B Add A + B to both sides: F - A - B + A + B = A - B + A + B which makes F = 2×A
Hint #2
Subtract A from both sides of eq.6: A + C - E - A = F - A - A which becomes C - E = F - 2×A Replace 2×A with F in the above equation: C - E = F - F which becomes C - E = 0 Add E to each side: C - E + E = 0 + E which means C = E
Hint #3
Substitute 2×A for F, and C for E in eq.4: 2×A - C = C - A Add A + C to both sides of the above equation: 2×A - C + A + C = C - A + A + C which simplifies to 3×A = 2×C Divide each side by 2: 3×A ÷ 2 = 2×C ÷ 2 which makes 1½×A = C which means 1½×A = E
Hint #4
In eq.5, replace E with 1½×A, and F with 2×A: B = 1½×A + 2×A which makes B = 3½×A
Hint #5
In eq.2, substitute 3½×A for B: A + 3½×A = D which makes 4½×A = D
Solution
Substitute 3½×A for B, 1½×A for C and E, 4½×A for D, and 2×A for F in eq.1: A + 3½×A + 1½×A + 4½×A + 1½×A + 2×A = 28 which simplifies to 14×A = 28 Divide each side by 14: 14×A ÷ 14 = 28 ÷ 14 which means A = 2 making B = 3½×A = 3½ × 2 = 7 C = E = 1½×A = 1½ × 2 = 3 D = 4½×A = 4½ × 2 = 9 F = 2×A = 2 × 2 = 4 and ABCDEF = 273934