Puzzle for November 23, 2023 ( )
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
Add D and E to both sides of eq.4: A - D + D + E = B - E + F + D + E which becomes eq.4a) A + E = B + D + F
Hint #2
In eq.4a, replace A + E with C + F (from eq.3): C + F = B + D + F Subtract F from each side of the equation above: C + F - F = B + D + F - F which becomes eq.4b) C = B + D
Hint #3
In eq.5, substitute (B + D) for C (from eq.4b): B - (B + D) + F = (B + D) + D + E - B which becomes B - B - D + F = 2×D + E which becomes -D + F = 2×D + E Add D to both sides of the above equation: -D + F + D = 2×D + E + D which becomes eq.5a) F = 3×D + E
Hint #4
eq.6 may be written as: D + E = (A + F) ÷ 2 Multiply both sides of the above equation by 2: 2 × (D + E) = 2 × (A + F) ÷ 2 which becomes eq.6a) 2×D + 2×E = A + F
Hint #5
In eq.6a, replace F with 3×D + E (from eq.5a): 2×D + 2×E = A + 3×D + E Subtract 2×D and E from both sides of the equation above: 2×D + 2×E - 2×D - E = A + 3×D + E - 2×D - E which becomes eq.6b) E = A + D
Hint #6
In eq.2, substitute E for A + D (from eq.6b): B + E = E Subtract E from each side of the equation above: B + E - E = E - E which makes B = 0
Hint #7
Substitute 0 for B in eq.4b: C = 0 + D which makes C = D
Hint #8
Substitute D for C, and 3×D + E for F (from eq.5a) in eq.3: D + 3×D + E = A + E which becomes 4×D + E = A + E Subtract E from each side of the above equation: 4×D + E - E = A + E - E which makes 4×D = A
Hint #9
Substitute 4×D for A in eq.6b: E = 4×D + D which makes E = 5×D
Hint #10
Substitute 5×D for E in eq.5a: F = 3×D + 5×D which makes F = 8×D
Solution
Substitute 4×D for A, 0 for B, D for C, 5×D for E, and 8×D for F in eq.1: 4×D + 0 + D + D + 5×D + 8×D = 19 which simplifies to 19×D = 19 Divide both sides of the above equation by 19: 19×D ÷ 19 = 19 ÷ 19 which means D = 1 making A = 4×D = 4 × 1 = 4 C = D = 1 E = 5×D = 5 × 1 = 5 F = 8×D = 8 × 1 = 8 and ABCDEF = 401158