Puzzle for February 6, 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.
* AB, BC, CD, and EF are 2-digit numbers (not A×B, B×C, C×D, or E×F).
Scratchpad
Help Area
Hint #1
eq.4 may be re-written as: E - F = C - (B + A) which is equivalent to E - F = C - (A + B) In the above equation, replace (A + B) with F (from eq.2): E - F = C - F Add F to both sides: E - F + F = C - F + F which makes E = C
Hint #2
eq.6 may be written as: 10×C + D - E + F = 10×E + F Add (E - F) to both sides of the equation above: 10×C + D - E + F + (E - F) = 10×E + F + (E - F) which simplifies to 10×C + D = 11×E Replace E with C: 10×C + D = 11×C Subtract 10×C from both sides: 10×C + D - 10×C = 11×C - 10×C which makes D = C
Hint #3
In eq.3, replace D with C: A - B = C - C which means A - B = 0 Add B to each side of the above equation: A - B + B = 0 + B which means A = B
Hint #4
eq.5 may be written as: A + 10×E + F - (10×B + C) = 10×A + B + D which is equivalent to A + 10×E + F - 10×B - C = 10×A + B + D Subtract both A and B from each side of the above equation: A + 10×E + F - 10×B - C - A - B = 10×A + B + D - A - B which becomes eq.5a) 10×E + F - 11×B - C = 9×A + D
Hint #5
Substitute C for D and E, and A for B in eq.5a: 10×C + F - 11×A - C = 9×A + C Add (11×A - C) to each side: 10×C + F - 11×A - C + (11×A - C) = 9×A + C + (11×A - C) which simplifies to 8×C + F = 20×A Subtract 8×C from both sides: 8×C + F - 8×C = 20×A - 8×C which becomes eq.5b) F = 20×A - 8×C
Hint #6
Substitute A for B, C for E, and (20×A - 8×C) for F (from eq.5b) in eq.4: C - (20×A - 8×C) = C - A - A which becomes C - 20×A + 8×C = C - 2×A Add (20×A - C) to both sides of the equation above: C - 20×A + 8×C + (20×A - C) = C - 2×A + (20×A - C) which simplifies to 8×C = 18×A Divide both sides by 8: 8×C ÷ 8 = 18×A ÷ 8 which makes C = 2¼×A which also makes D = E = C = 2¼×A
Hint #7
Substitute 2¼×A for C in eq.5b: F = 20×A - 8×(2¼×A) which becomes F = 20×A - 18×A which makes F = 2×A
Solution
Substitute A for B, 2¼×A for C and D and E, and 2×A for F in eq.1: A + A + 2¼×A + 2¼×A + 2¼×A + 2×A = 43 which simplifies to 10¾×A = 43 Divide both sides of the equation above by 10¾: 10¾×A ÷ 10¾ = 43 ÷ 10¾ which means A = 4 making B = A = 4 C = D = E = 2¼×A = 2¼ × 4 = 9 F = 2×A = 2 × 4 = 8 and ABCDEF = 449998