Puzzle for March 19, 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 DE are 2-digit numbers (not C×D or D×E).
Scratchpad
Help Area
Hint #1
In eq.5, replace C with D + E (from eq.3): A + D - E = B + D + E Add (E - D) to both sides of the above equation: A + D - E + (E - D) = B + D + E + (E - D) which becomes eq.5a) A = B + 2×E
Hint #2
In eq.2, replace A with B + 2×E (from eq.5a): B + 2×E = B + E + F Subtract both B and E from each side of the above equation: B + 2×E - B - E = B + E + F - B - E which makes E = F
Hint #3
Subtract both C and D from each side of eq.5: A + D - E - C - D = B + C - C - D which becomes A - E - C = B - D In the equation above, substitute A - C + F for B - D (from eq.4): A - E - C = A - C + F Add (C - A) to both sides: A - E - C + (C - A) = A - C + F + (C - A) which simplifies to -E = F Substitute E for F, and add E to both sides: -E + E = E + E which makes 0 = 2×E which means 0 = E which also makes F = E = 0
Hint #4
Substitute 0 for E in eq.3: C = D + 0 C = D
Hint #5
Substitute 0 for E and F in eq.2: A = B + 0 + 0 which makes A = B
Hint #6
eq.6 can be written as: 10×D + E + A = 10×C + D - A - B + F Substitute A for B, C for D, and 0 for both E and F in the above equation: 10×C + 0 + A = 10×C + C - A - A + 0 which becomes 10×C + A = 11×C - 2×A Add (2×A - 10×C) to both sides: 10×C + A + (2×A - 10×C) = 11×C - 2×A + (2×A - 10×C) which makes 3×A = C which also makes D = C = 3×A
Solution
Substitute A for B, 3×A for C and D, and 0 for E and F in eq.1: A + A + 3×A + 3×A + 0 + 0 = 8 which simplifies to 8×A = 8 Divide both sides by 8: 8×A ÷ 8 = 8 ÷ 8 which means A = 1 making B = A = 1 C = D = 3×A = 3 × 1 = 3 and ABCDEF = 113300