Puzzle for September 16, 2022 ( )
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 F to both sides of eq.3: E + F + F = D - F + F which becomes eq.3a) E + 2×F = D Add D to both sides of eq.5: B + C - D + D = A + D + E + F + D which becomes eq.5a) B + C = A + 2×D + E + F
Hint #2
In eq.5a, substitute (E + 2×F) for D (from eq.3a): B + C = A + 2×(E + 2×F) + E + F which becomes B + C = A + 2×E + 4×F + E + F which becomes eq.5b) B + C = A + 3×E + 5×F
Hint #3
eq.6 may be written as: D = (B + C + E + F) ÷ 4 Multiply both sides of the above equation by 4: 4 × D = 4 × (B + C + E + F) ÷ 4 which becomes eq.6a) 4×D = B + C + E + F
Hint #4
In eq.6a, substitute (E + 2×F) for D (from eq.3a), and A + 3×E + 5×F for B + C (from eq.5b): 4×(E + 2×F) = A + 3×E + 5×F + E + F which becomes 4×E + 8×F = A + 4×E + 6×F Subtract 4×E and 6×F from both sides of the equation above: 4×E + 8×F - 4×E - 6×F = A + 4×E + 6×F - 4×E - 6×F which simplifies to 2×F = A
Hint #5
In eq.4, replace A with 2×F: C - 2×F = 2×F + D - F which becomes C - 2×F = F + D Add 2×F to both sides of the equation above: C - 2×F + 2×F = F + D + 2×F which becomes eq.4a) C = 3×F + D
Hint #6
In eq.2, replace A with 2×F, and C with 3×F + D (from eq.4a): D - 2×F = 3×F + D - D which becomes D - 2×F = 3×F Add 2×F to both sides of the above equation: D - 2×F + 2×F = 3×F + 2×F which makes D = 5×F
Hint #7
Substitute 5×F for D in eq.3a: E + 2×F = 5×F Subtract 2×F from each side of the above equation: E + 2×F - 2×F = 5×F - 2×F which makes E = 3×F
Hint #8
Substitute 5×F for D in eq.4a: C = 3×F + 5×F which makes C = 8×F
Hint #9
Substitute 8×F for C, 2×F for A, and (3×F) for E in eq.5b: B + 8×F = 2×F + 3×(3×F) + 5×F which becomes B + 8×F = 2×F + 9×F + 5×F which becomes B + 8×F = 16×F Subtract 8×F from both sides of the above equation: B + 8×F - 8×F = 16×F - 8×F which makes B = 8×F
Solution
Substitute 2×F for A, 8×F for B and C, 5×F for D, and 3×F for E in eq.1: 2×F + 8×F + 8×F + 5×F + 3×F + F = 27 which simplifies to 27×F = 27 Divide both sides of the above equation by 27: 27×F ÷ 27 = 27 ÷ 27 which means F = 1 making A = 2×F = 2 × 1 = 2 B = C = 8×F = 8 × 1 = 8 D = 5×F = 5 × 1 = 5 E = 3×F = 3 × 1 = 3 and ABCDEF = 288531