Puzzle for December 9, 2023 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit positive integer.
Scratchpad
Help Area
Hint #1
In eq.5, substitute (C + D) for F (from eq.2): E - B - C - D = A - (C + D) which becomes E - B - C - D = A - C - D Add B, C, and D to both sides of the above equation: E - B - C - D + B + C + D = A - C - D + B + C + D which simplifies to eq.5a) E = A + B
Hint #2
In eq.3, replace E with A + B (from eq.5a): A + F = C + A + B Subtract A from each side of the equation above: A + F - A = C + A + B - A which becomes eq.3a) F = C + B
Hint #3
In eq.2, replace F with C + B (from eq.3a): C + B = C + D Subtract C from each side of the above equation: C + B - C = C + D - C which makes B = D
Hint #4
In eq.4, substitute B for D: C + B + F = A + B + E Subtract B from both sides of the above equation: C + B + F - B = A + B + E - B which becomes eq.4a) C + F = A + E
Hint #5
Subtract the left and right sides of eq.4a from the left and right sides of eq.3, respectively: A + F - (C + F) = C + E - (A + E) which becomes A + F - C - F = C + E - A - E which becomes A - C = C - A Add C and A to both sides of the above equation: A - C + C + A = C - A + C + A which becomes 2×A = 2×C Divide both sides by 2: 2×A ÷ 2 = 2×C ÷ 2 which makes A = C
Hint #6
Substitute A for C in eq.4a: A + F = A + E Subtract A from both sides of the equation above: A + F - A = A + E - A which makes F = E
Hint #7
Substitute E for F in eq.2: eq.2a) E = C + D
Hint #8
Substitute E for A + B (from eq.5a), E for C + D (from eq.2a), and E for F in eq.1: E + E + E + E = 28 which makes 4×E = 28 Divide both sides of the above equation by 4: 4×E ÷ 4 = 28 ÷ 4 which makes E = 7 and also makes F = E = 7
Hint #9
Substitute 7 for F in eq.2: 7 = C + D Subtract D from each side of the equation above: 7 - D = C + D - D which makes 7 - D = C and also makes eq.2b) 7 - D = C = A
Hint #10
Substitute D for B, 7 for E, and (7 - D) for A and C (from eq.2b) in eq.6: D + 7 = (7 - D) + ((7 - D) × D) which becomes D + 7 = 7 - D + 7×D - D² which becomes D + 7 = 7 + 6×D - D² Subtract 7 and 6×D from both sides of the above equation: D + 7 - 7 - 6×D = 7 + 6×D - D² - 7 - 6×D which simplifies to -5×D = -D² Divide both sides by (-D): -5×D ÷ (-D) = -D² ÷ (-D) which makes 5 = D
Solution
Since D = 5, then: A = C = 7 - D = 7 - 5 = 2 (from eq.2b) B = D = 5 and ABCDEF = 252577