Puzzle for December 30, 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
In eq.5, replace A with B + D (from eq.3): D + F = B + D + B which becomes D + F = 2×B + D Subtract D from each side of the equation above: D + F – D = 2×B + D – D which makes F = 2×B
Hint #2
In eq.4, replace A + D with E (from eq.6): E + F = C + E Subtract E from both sides of the above equation: E + F – E = C + E – E which makes F = C and also means eq.4a) F = C = 2×B
Hint #3
In eq.2, substitute 2×B for F, and B + D for A (from eq.3): 2×B – D = B + D + D which becomes 2×B – D = B + 2×D In the above equation, add D to both sides, and subtract B from each side: 2×B – D + D – B = B + 2×D + D – B which makes B = 3×D
Hint #4
Substitute (3×D) for B in eq.4a: F = C = 2×(3×D) which makes F = C = 6×D
Hint #5
Substitute 3×D for B in eq.3: 3×D + D = A which makes 4×D = A
Hint #6
Substitute 4×D for A in eq.6: E = 4×D + D which makes E = 5×D
Solution
Substitute 4×D for A, 3×D for B, 6×D for C and F, and 5×D for E in eq.1: 4×D + 3×D + 6×D + D + 5×D + 6×D = 25 which simplifies to 25×D = 25 Divide both sides of the above equation by 25: 25×D ÷ 25 = 25 ÷ 25 which means D = 1 making A = 4×D = 4 × 1 = 4 B = 3×D = 3 × 1 = 3 C = F = 6×D = 6 × 1 = 6 E = 5×D = 5 × 1 = 5 and ABCDEF = 436156