Puzzle for September 23, 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
In eq.5, replace C + E with B + D - E (from eq.4): B + D + F = A - B + B + D - E which becomes B + D + F = A + D - E In the above equation, subtract D from both sides, and add E to both sides: B + D + F - D + E = A + D - E - D + E which becomes eq.5a) B + F + E = A
Hint #2
In eq.3, replace A with B + F + E (from eq.5a): C + F = B + F + E + B + E which becomes C + F = 2×B + F + 2×E Subtract F from each side of the equation above: C + F - F = 2×B + F + 2×E - F which becomes eq.3a) C = 2×B + 2×E
Hint #3
In eq.2, substitute 2×B + 2×E for C (from eq.3a): E - B = 2×B + 2×E - E which becomes E - B = 2×B + E In the above equation, add B to both sides, and subtract E from both sides: E - B + B - E = 2×B + E + B - E which makes 0 = 3×B which means 0 = B
Hint #4
Substitute 0 for B in eq.3a: C = 2×0 + 2×E which means C = 0 + 2×E which makes C = 2×E
Hint #5
Substitute 2×E for C, and 0 for B in eq.4: 2×E + E = 0 + D - E which becomes 3×E = D - E Add E to both sides of the equation above: 3×E + E = D - E + E which makes 4×E = D
Hint #6
Substitute 0 for B in eq.5a: 0 + F + E = A which becomes eq.5b) F + E = A
Hint #7
eq.6 may be written as: F = (A + C + D) ÷ 3 Multiply both sides of the above equation by 3: 3 × F = 3 × (A + C + D) ÷ 3 which becomes eq.6a) 3×F = A + C + D
Hint #8
Substitute F + E for A (from eq.5b), 2×E for C, and 4×E for D in eq.6a: 3×F = F + E + 2×E + 4×E which becomes 3×F = F + 7×E Subtract F from each side of the equation above: 3×F - F = F + 7×E - F which becomes 2×F = 7×E Divide both sides by 2: 2×F ÷ 2 = 7×E ÷ 2 which makes F = 3½×E
Hint #9
Substitute 3½×E for F in eq.5b: 3½×E + E = A which makes 4½×E = A
Solution
Substitute 4½×E for A, 0 for B, 2×E for C, 4×E for D, and 3½×E for F in eq.1: 4½×E + 0 + 2×E + 4×E + E + 3½×E = 30 which simplifies to 15×E = 30 Divide both sides of the above equation by 15: 15×E ÷ 15 = 30 ÷ 15 which means E = 2 making A = 4½×E = 4½ × 2 = 9 C = 2×E = 2 × 2 = 4 D = 4×E = 4 × 2 = 8 F = 3½×E = 3½ × 2 = 7 and ABCDEF = 904827