Puzzle for February 22, 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.2, replace C with D + E (from eq.3): D + E + D = E + F which becomes 2×D + E = E + F Subtract E from each side of the above equation: 2×D + E – E = E + F – E which makes 2×D = F
Hint #2
In eq.4, replace F with 2×D: E = D + 2×D which makes E = 3×D
Hint #3
In eq.3, substitute 3×D for E: D + 3×D = C which makes 4×D = C
Hint #4
Substitute 4×D for C in eq.6: B + D = 4×D – B In the above equation, subtract D from both sides, and add B to both sides: B + D – D + B = 4×D – B – D + B which becomes 2×B = 3×D Divide both sides by 2: 2×B ÷ 2 = 3×D ÷ 2 which makes B = 1½×D
Hint #5
Substitute 1½×D for B, and 3×D for E in eq.5: A = 1½×D + 3×D which makes A = 4½×D
Solution
Substitute 4½×D for A, 1½×D for B, 4×D for C, 3×D for E, and 2×D for F in eq.1: 4½×D + 1½×D + 4×D + D + 3×D + 2×D = 32 which simplifies to 16×D = 32 Divide both sides of the equation above by 16: 16×D ÷ 16 = 32 ÷ 16 which means D = 2 making A = 4½×D = 4½ × 2 = 9 B = 1½×D = 1½ × 2 = 3 C = 4×D = 4 × 2 = 8 E = 3×D = 3 × 2 = 6 F = 2×D = 2 × 2 = 4 and ABCDEF = 938264