Puzzle for October 25, 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
In eq.6, replace A + B with C (from eq.4): C – D + F = C + D In the above equation, subtract C from both sides, and add D to both sides: C – D + F – C + D = C + D – C + D which makes F = 2×D
Hint #2
In eq.2, replace F with 2×D: 2×D = B + D Subtract D from each side of the equation above: 2×D – D = B + D – D which makes D = B
Hint #3
In eq.5, substitute D for B: E – D = D + D – E which becomes E – D = 2×D – E Add D and E to both sides of the above equation: E – D + D + E = 2×D – E + D + E which makes 2×E = 3×D Divide both sides by 2: 2×E ÷ 2 = 3×D ÷ 2 which makes E = 1½×D
Hint #4
Substitute 1½×D for E in eq.3: A = D + 1½×D which makes A = 2½×D
Hint #5
Substitute 2½×D for A, and D for B in eq.4: C = 2½×D + D which makes C = 3½×D
Solution
Substitute 2½×D for A, D for B, 3½×D for C, 1½×D for E, and 2×D for F in eq.1: 2½×D + D + 3½×D + D + 1½×D + 2×D = 23 which simplifies to 11½×D = 23 Divide both sides of the above equation by 11½: 11½×D ÷ 11½ = 23 ÷ 11½ which means D = 2 making A = 2½×D = 2½ × 2 = 5 B = D = 2 C = 3½×D = 3½ × 2 = 7 E = 1½×D = 1½ × 2 = 3 F = 2×D = 2 × 2 = 4 and ABCDEF = 527234