Puzzle for December 1, 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.
* AB, BC, and CD are 2-digit numbers (not A×B, B×C, or C×D).
Scratchpad
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Hint #1
eq.6 may be written as: 10×A + B + 10×C + D = 10×B + C + E Subtract both B and C from each side of the equation above: 10×A + B + 10×C + D - B - C = 10×B + C + E - B - C which becomes 10×A + 9×C + D = 9×B + E Substitute A + D for E (from eq.2): 10×A + 9×C + D = 9×B + A + D Subtract A and D from each side: 10×A + 9×C + D - A - D = 9×B + A + D - A - D which becomes 9×A + 9×C = 9×B Divide both sides by 9: (9×A + 9×C) ÷ 9 = 9×B ÷ 9 which means eq.6a) A + C = B
Hint #2
Substitute A + C for B (from eq.6a), and A + D for E (from eq.2) in eq.3: A + C + C = D + A + D which is the same as A + 2×C = A + 2×D Subtract A from each side of the above equation: A + 2×C - A = A + 2×D - A which simplifies to 2×C = 2×D Divide both sides by 2: 2×C ÷ 2 = 2×D ÷ 2 which makes C = D
Hint #3
Substitute C for D in eq.3: B + C = C + E Subtract C from both sides: B + C - C = C + E - C which makes B = E
Hint #4
Add A + F to each side of eq.5: A + F + A + F = C + E - A - F + A + F which becomes 2×A + 2×F = C + E Substitute D for C, and A + D for E (from eq.2) in the equation above: 2×A + 2×F = D + A + D Subtract A from each side: 2×A + 2×F - A = D + A + D - A which becomes eq.5a) A + 2×F = 2×D
Hint #5
Substitute A + C for B (from eq.6a) in eq.4: C + D = A + A + C - F Subtract C from each side of the equation above: C + D - C = A + A + C - F - C which becomes D = 2×A - F Substitute (2×A - F) for D in eq.5a: A + 2×F = 2×(2×A - F) which becomes A + 2×F = 4×A - 2×F Add (2×F - A) to each side: A + 2×F + (2×F - A) = 4×A - 2×F + (2×F - A) which simplifies to 4×F = 3×A Divide both sides by 4: 4×F ÷ 4 = 3×A ÷ 4 which makes F = ¾×A
Hint #6
Substitute (¾×A) for F in eq.5a: A + 2×(¾×A) = 2×D which becomes A + 1½×A = 2×D which means 2½×A = 2×D Divide both sides by 2: 2½×A ÷ 2 = 2×D ÷ 2 which makes 1¼×A = D
Hint #7
Substitute 1¼×A for D in eq.2: A + 1¼×A = E which makes 2¼×A = E
Solution
Substitute 2¼×A for E and B, 1¼×A for D and C, and ¾×A for F in eq.1: A + 2¼×A + 1¼×A + 1¼×A + 2¼×A + ¾×A = 35 which simplifies to 8¾×A = 35 Divide both sides by 8¾: 8¾×A ÷ 8¾ = 35 ÷ 8¾ which means A = 4 making B = E = 2¼×A = 2¼ × 4 = 9 C = D = 1¼×A = 1¼ × 4 = 5 F = ¾×A = ¾ × 4 = 3 and ABCDEF = 495593