Puzzle for January 10, 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.
* BC, CD, and EF are 2-digit numbers (not B×C, C×D, or E×F).
Scratchpad
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Hint #1
In eq.3, replace E + F with A + C (from eq.2): C + A + C = D which becomes A + 2×C = D In eq.4, replace D with A + 2×C: A + 2×C - B = A + B - C Add (B + C - A) to each side of the above equation: A + 2×C - B + (B + C - A) = A + B - C + (B + C - A) which simplifies to 3×C = 2×B Divide both sides by 2: 3×C ÷ 2 = 2×B ÷ 2 which means 1½×C = B
Hint #2
eq.6 can be written as: 10×B + C - D = 10×C + D Add (D - C) to both sides of the above equation: 10×B + C - D + (D - C) = 10×C + D + (D - C) which becomes 10×B = 9×C + 2×D Replace B with (1½×C) in the above equation: 10×(1½×C) = 9×C + 2×D which becomes 15×C = 9×C + 2×D Subtract 9×C from each side: 15×C - 9×C = 9×C + 2×D - 9×C which simplifies to 6×C = 2×D Divide both sides by 2: 6×C ÷ 2 = 2×D ÷ 2 which means 3×C = D
Hint #3
Substitute 1½×C for B, and 3×C for D in eq.4: 3×C - 1½×C = A + 1½×C - C which becomes 1½×C = A + ½×C Subtract ½×C from both sides of the equation above: 1½×C - ½×C = A + ½×C - ½×C which makes C = A
Hint #4
Substitute 3×C for D, and subtract C from each side of eq.3: C + E + F - C = 3×C - C which becomes eq.3a) E + F = 2×C eq.5 may be written as: 10×C + D - A = 10×E + F + C which can also be written as 10×C + D - A = 9×E + E + F + C Substitute C for A, 3×C for D, and 2×C for E + F (from eq.3a) in the above equation: 10×C + 3×C - C = 9×E + 2×C + C which becomes 12×C = 9×E + 3×C Subtract 3×C from each side: 12×C - 3×C = 9×E + 3×C - 3×C which simplifies to 9×C = 9×E Divide both sides by 9: 9×C ÷ 9 = 9×E ÷ 9 which means C = E
Hint #5
Substitute C for both A and E in eq.2: C + C = C + F Subtract C from each side of the equation above: C + C - C = C + F - C which means C = F
Solution
Substitute C for A and E and F, 1½×C for B, and 3×C for D in eq.1: C + 1½×C + C + 3×C + C + C = 17 which becomes 8½×C = 17 Divide both sides by 8½: 8½×C ÷ 8½ = 17 ÷ 8½ which makes C = 2 making A = C = E = F = 2 B = 1½×C = 1½ × 2 = 3 D = 3×C = 3 × 2 = 6 and ABCDEF = 232622