Puzzle for December 20, 2018 ( )
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.
* BC and EF are 2-digit numbers (not B×C or E×F).
Scratchpad
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Hint #1
Subtract E from both sides of eq.5: B - E = D + E - E which becomes B - E = D In eq.2, replace D with (B - E): F - (B - E) = C + E which is equivalent to F - B + E = C + E Subtract E from both sides of the above equation: F - B + E - E = C + E - E which reduces to eq.5a) F - B = C In eq.3, replace C with F - B: A - B = F - B Add B to each side: A - B + B = F - B + B which means A = F
Hint #2
Add B to both sides of eq.5a: F - B + B = C + B which becomes eq.3a) F = C + B
Hint #3
eq.6 may be written as: 10×B + C - (10×E + F) = E - D which is equivalent to 10×B + C - 10×E - F = E - D Add 10×E to both sides of the above equation: 10×B + C - 10×E - F + 10×E = E - D + 10×E which becomes 10×B + C - F = 11×E - D Substitute (C + B) for F (from eq.3a): 10×B + C - (C + B) = 11×E - D which is equivalent to 10×B + C - C - B = 11×E - D which becomes eq.6a) 9×B = 11×E - D
Hint #4
In eq.6a, substitute (D + E) for B (from eq.5): 9×(D + E) = 11×E - D which is the same as 9×D + 9×E = 11×E - D Add (D - 9×E) to each side: 9×D + 9×E + (D - 9×E) = 11×E - D + (D - 9×E) which simplifies to 10×D = 2×E Divide both sides by 2: 10×D ÷ 2 = 2×E ÷ 2 which means 5×D = E
Hint #5
Substitute 5×D for E in eq.5: B = D + 5×D which makes B = 6×D
Hint #6
In eq.4, replace E with 5×D: 5×D - D = A - 5×D Add 5×D to each side: 5×D - D + 5×D = A - 5×D + 5×D which means 9×D = A
Hint #7
Substitute 9×D for A, and 6×D for B in eq.3: 9×D - 6×D = C which makes 3×D = C
Solution
Substitute 9×D for A and F, 6×D for B, 3×D for C, and 5×D for E in eq.1: 9×D + 6×D + 3×D + D + 5×D + 9×D = 33 which becomes 33×D = 33 Divide both sides by 33: 33×D ÷ 33 = 33 ÷ 33 which makes D = 1 making A = F = 9×D = 9 × 1 = 9 B = 6×D = 6 × 1 = 6 C = 3×D = 3 × 1 = 3 E = 5×D = 5 × 1 = 5 and ABCDEF = 963159