Puzzle for July 11, 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.3, replace C with A - D (from eq.6): F = A - D + D which makes F = A In eq.3, replace F with A: eq.3a) A = C + D
Hint #2
In eq.2, replace A with F: E + F = F + B + C Subtract F from both sides of the above equation: E + F - F = F + B + C - F which becomes eq.2a) E = B + C
Hint #3
In eq.4, substitute C + D for A (from eq.3a) and for F (from eq.3): C + D + C + D = C - D + E which becomes 2×C + 2×D = C - D + E In the above equation, subtract C from both sides, and add D to both sides: 2×C + 2×D - C + D = C - D + E - C + D which simplifies to eq.4a) C + 3×D = E
Hint #4
Substitute C + 3×D for E (from eq.4a) in eq.2a: C + 3×D = B + C Subtract C from both sides of the above equation: C + 3×D - C = B + C - C which makes 3×D = B
Hint #5
In eq.5, substitute (B + C) for E (from eq.2a): B - C = F - (B + C) which is the same as B - C = F - B - C Add B and C to both sides of the above equation: B - C + B + C = F - B - C + B + C which simplifies to eq.2b) 2×B = F
Hint #6
Substitute (3×D) for B in eq.2b: 2×(3×D) = F which makes 6×D = F and also makes A = F = 6×D
Hint #7
Substitute 6×D for F in eq.3: 6×D = C + D Subtract D from both sides: 6×D - D = C + D - D which makes 5×D = C
Hint #8
Substitute 3×D for B, and 5×D for C in eq.2a: E = 3×D + 5×D which makes E = 8×D
Solution
Substitute 6×D for A and F, 3×D for B, 5×D for C, and 8×D for E in eq.1: 6×D + 3×D + 5×D + D + 8×D + 6×D = 29 which simplifies to 29×D = 29 Divide both sides of the above equation by 29: 29×D ÷ 29 = 29 ÷ 29 which means D = 1 making A = F = 6×D = 6×1 = 6 B = 3×D = 3×1 = 3 C = 5×D = 5×1 = 5 E = 8×D = 8×1 = 8 and ABCDEF = 635186