Puzzle for July 5, 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.
Scratchpad
Help Area
Hint #1
Add A and F to both sides of eq.3: D + A + F = E - F - A + A + F which becomes D + A + F = E which may be written as eq.3a) A + D + F = E
Hint #2
Add E and F to both sides of eq.5: B - C - F + E + F = A + D - E + E + F which becomes B - C + E = A + D + F In the above equation, replace A + D + F with E (from eq.3a): B - C + E = E Subtract E from both sides, and add C to both sides: B - C + E - E + C = E - E + C which simplifies to B = C
Hint #3
In eq.6, replace C with B: F - B = D - B Add B to both sides of the above equation: F - B + B = D - B + B which makes F = D
Hint #4
In eq.4, substitute B for C, and D for F: B - D = E + D - B Add B and D to both sides of the above equation: B - D + B + D = E + D - B + B + D eq.4a) 2×B = E + 2×D
Hint #5
In eq.3a, substitute B + D for A (from eq.2), and D for F: B + D + D + D = E which becomes eq.5a) B + 3×D = E
Hint #6
Substitute B + 3×D for E (from eq.5a) in eq.4a: 2×B = B + 3×D + 2×D which becomes 2×B = B + 5×D Subtract B from both sides of the above equation: 2×B - B = B + 5×D - B which makes B = 5×D and which also makes C = B = 5×D
Hint #7
Substitute 5×D for B in eq.2: A = 5×D + D which makes A = 6×D
Hint #8
Substitute 5×D for B in eq.5a: 5×D + 3×D = E which makes 8×D = E
Solution
Substitute 6×D for A, 5×D for B and C, 8×D for E, and D for F in eq.1: 6×D + 5×D + 5×D + D + 8×D + D = 26 which simplifies to 26×D = 26 Divide both sides of the above equation by 26: 26×D ÷ 26 = 26 ÷ 26 which means D = 1 making A = 6×D = 6×1 = 6 B = C = 5×D = 5×1 = 5 E = 8×D = 8×1 = 8 F = D = 1 and ABCDEF = 655181