Puzzle for September 18, 2021 ( )
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
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Hint #1
In eq.3, replace F with C + E (from eq.2): B + C + E = A + E Subtract E from both sides of the above equation: B + C + E – E = A + E – E which becomes eq.3a) B + C = A
Hint #2
In eq.5, replace A with B + C (from eq.3a): B + C + C + D – E = B + E which becomes B + 2×C + D – E = B + E In the above equation, subtract B from both sides, and add E to both sides: B + 2×C + D – E – B + E = B + E – B + E which becomes eq.5a) 2×C + D = 2×E
Hint #3
Add D to both sides of eq.4: D + E + D = A – D + D which becomes eq.4a) 2×D + E = A In eq.5, substitute 2×D + E for A (from eq.4a): 2×D + E + C + D – E = B + E which becomes 3×D + C = B + E Subtract 3×D from each side of the above equation: 3×D + C – 3×D = B + E – 3×D which becomes eq.5b) C = B + E – 3×D
Hint #4
Substitute (B + E – 3×D) for C (from eq.5b) in eq.5a: 2×(B + E – 3×D) + D = 2×E which becomes 2×B + 2×E – 6×D + D = 2×E which becomes 2×B + 2×E – 5×D = 2×E In the above equation, subtract 2×E from both sides, and add 5×D to both sides: 2×B + 2×E – 5×D – 2×E + 5×D = 2×E – 2×E + 5×D which simplifies to 2×B = 5×D Divide both sides by 2: 2×B ÷ 2 = 5×D ÷ 2 which makes B = 2½×D
Hint #5
Substitute 2½×D for B in eq.5b: C = 2½×D + E – 3×D which becomes C = –½×D + E which is equivalent to eq.5c) C = E – ½×D
Hint #6
Substitute E – ½×D for C (from eq.5b) into eq.2: F = E – ½×D + E which becomes eq.2a) F = 2×E – ½×D
Hint #7
eq.6 may be written as: B = (C + D + F) ÷ 3 Multiply both sides of the above equation by 3: B × 3 = ((C + D + F) ÷ 3) × 3 which becomes B × 3 = C + D + F which may be written as eq.6a) 3×B = C + D + F
Hint #8
In eq.6a, substitute 2½×D for B, E – ½×D for C (from eq.5c), and 2×E – ½×D for F (from eq.2a): 3×(2½×D) = E – ½×D + D + 2×E – ½×D which becomes 7½×D = 3×E Divide both sides of the above equation by 3: 7½×D ÷ 3 = 3×E ÷ 3 which becomes 2½×D = E
Hint #9
Substitute (2½×D) for E in eq.2a: F = 2×(2½×D) – ½×D which becomes F = 5×D – ½×D which makes F = 4½×D
Hint #10
Substitute 2½×D for E in eq.5c: C = 2½×D – ½×D which makes C = 2×D
Hint #11
Substitute 2½×D for E in eq.4a: 2×D + 2½×D = A which makes 4½×D = A
Solution
Substitute 4½×D for A and F, 2½×D for B and E, and 2×D for C in eq.1: 4½×D + 2½×D + 2×D + D + 2½×D + 4½×D = 34 which simplifies to 17×D = 34 Divide both sides of the above equation by 17: 17×D ÷ 17 = 34 ÷ 17 which means D = 2 making A = F = 4½×D = 4½ × 2 = 9 B = E = 2½×D = 2½ × 2 = 5 C = 2×D = 2 × 2 = 4 and ABCDEF = 954259