Puzzle for January 17, 2019 ( )
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.
* CD and EF are 2-digit numbers (not C×D or E×F).
Scratchpad
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Hint #1
Subtract both B and C from each side of eq.2: A + B - B - C = C + D - B - C which becomes A - C = D - B In eq.5, replace A - C with D - B: D - B - F = F - E Add E + F to both sides of the above equation: D - B - F + E + F = F - E + E + F which becomes eq.5a) D - B + E = 2×F
Hint #2
Add F to both sides of eq.4: B + D - F + F = F + F which becomes B + D = 2×F In eq.5a, replace 2×F with B + D: D - B + E = B + D Add (B - D) to both sides of the above equation: D - B + E + (B - D) = B + D + (B - D) which simplifies to E = 2×B
Hint #3
In eq.3, replace E with 2×B: B + C = 2×B Subtract B from both sides of the equation above: B + C - B = 2×B - B which means C = B
Hint #4
In eq.2, substitute B for C: A + B = B + D Subtract B from each side of the equation above: A + B - B = B + D - B which means A = D
Hint #5
eq.6 may be written as: 10×E + F - (10×C + D) = A + D - E + F which is the same as 10×E + F - 10×C - D = A + D - E + F Add (D + E - F) to both sides of the above equation: 10×E + F - 10×C - D + (D + E - F) = A + D - E + F + (D + E - F) which simplifies to 11×E - 10×C = A + 2×D Substitute A for D, B for C, and (2×B) for E: 11×(2×B) - 10×B = A + 2×A which becomes 22×B - 10×B = 3×A which becomes 12×B = 3×A Divide both sides by 3: 12×B ÷ 3 = 3×A ÷ 3 which means 4×B = A
Hint #6
Substitute 4×B for A, B for C, and 2×B for E in eq.5: 4×B - B - F = F - 2×B Add 2×B + F to each side of the equation above: 4×B - B - F + 2×B + F = F - 2×B + 2×B + F which simplifies to 5×B = 2×F Divide both sides by 2: 5×B ÷ 2 = 2×F ÷ 2 which makes 2½×B = F
Solution
Substitute 4×B for A and D, B for C, 2×B for E, and 2½×B for F in eq.1: 4×B + B + B + 4×B + 2×B + 2½×B = 29 which simplifies to 14½×B = 29 Divide both sides by 14½: 14½×B ÷ 14½ = 29 ÷ 14½ which makes B = 2 making A = D = 4×B = 4 × 2 = 8 C = B = 2 E = 2×B = 2 × 2 = 4 F = 2½×B = 2½ × 2 = 5 and ABCDEF = 822845