Puzzle for January 21, 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
Add C and F to both sides of eq.6: A – C + E – F + C + F = C + F + C + F which becomes A + E = 2×C + 2×F which may be written as eq.6a) A + E = 2×(C + F)
Hint #2
In eq.6a, replace C + F with D + E (from eq.4): A + E = 2×(D + E) which becomes A + E = 2×D + 2×E Subtract E from each side: A + E – E = 2×D + 2×E – E which becomes eq.6b) A = 2×D + E
Hint #3
In eq.3, replace A with 2×D + E (from eq.6b): D + F = 2×D + E Subtract D from each side of the equation above: D + F – D = 2×D + E – D which becomes eq.3a) F = D + E
Hint #4
In eq.4, substitute F for D + E (from eq.3a): C + F = F Subtract F from each side of the equation above: C + F – F = F – F which makes C = 0
Hint #5
Substitute 2×D + E for A (from eq.6b), and 0 for C in eq.2: B + E = 2×D + E + 0 + D which becomes B + E = 3×D + E Subtract E from both sides of the equation above: B + E – E = 3×D + E – E which makes B = 3×D
Hint #6
Substitute D + E for F (from eq.3a), 3×D for B, and 0 for C in eq.5: E + D + E – (3×D – D) = 3×D + 0 + D which becomes 2×E + D – 3×D + D = 4×D which becomes 2×E – D = 4×D Add D to both sides of the above equation: 2×E – D + D = 4×D + D which becomes 2×E = 5×D Divide both sides by 2: 2×E ÷ 2 = 5×D ÷ 2 which makes E = 2½×D
Hint #7
Substitute 2½×D for E in eq.3a: F = D + 2½×D which makes F = 3½×D
Hint #8
Substitute 2½×D for E in eq.6b: A = 2×D + 2½×D which makes A = 4½×D
Solution
Substitute 4½×D for A, 3×D for B, 0 for C, 2½×D for E, and 3½×D for F in eq.1: 4½×D + 3×D + 0 + D + 2½×D + 3½×D = 29 which simplifies to 14½×D = 29 Divide both sides of the equation above by 14½: 14½×D ÷ 14½ = 29 ÷ 14½ which means D = 2 making A = 4½×D = 4½ × 2 = 9 B = 3×D = 3 × 2 = 6 E = 2½×D = 2½ × 2 = 5 F = 3½×D = 3½ × 2 = 7 and ABCDEF = 960257