Puzzle for September 21, 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 non-negative integer.
* "F!" means F-factorial.
Scratchpad
Help Area
Hint #1
In eq.4, replace C + D with E + F (from eq.3): A + E = E + F + F which becomes A + E = E + 2×F Subtract E from both sides of the above equation: A + E - E = E + 2×F - E which makes A = 2×F
Hint #2
eq.4 may be written as: A + E = C + F + D In the above equation, replace C + F with A + D (from eq.1): A + E = A + D + D which becomes A + E = A + 2×D Subtract A from each side of the equation above: A + E - A = A + 2×D - A which makes eq.4a) E = 2×D
Hint #3
In eq.5, substitute 2×D for E: B ÷ F = 2×D ÷ D which becomes B ÷ F = 2 Multiply both sides of the above equation by F: (B ÷ F) × F = 2×F which makes B = 2×F
Hint #4
Substitute 2×F for B in eq.2: 2×F - E = E - F Add E and F to both sides of the equation above: 2×F - E + E + F = E - F + E + F which makes 3×F = 2×E Divide both sides by 2: 3×F ÷ 2 = 2×E ÷ 2 which makes 1½×F = E
Hint #5
Substitute 1½×F for E in eq.4a: 1½×F = 2×D Divide both sides of the above equation by 2: 1½×F ÷ 2 = 2×D ÷ 2 which makes ¾×F = D
Hint #6
Substitute ¾×F for D, and 1½×F for E in eq.3: C + ¾×F = 1½×F + F which becomes C + ¾×F = 2½×F Subtract ¾×F from each side of the above equation: C + ¾×F - ¾×F = 2½×F - ¾×F which makes C = 1¾×F
Hint #7
Substitute 2×F for B, 1¾×F for C, ¾×F for D, and 1½×F for E in eq.6: F! = 2×F + 1¾×F + ¾×F + 1½×F which becomes F! = 6×F The above equation may be written as: 1 × 2 × 3 × ... × (F – 2) × (F – 1) × F = 6×F which is the same as (F – 1)! × F = 6×F Since F ≠ 0 (from eq.5), divide both sides by F: (F – 1)! × F ÷ F = 6×F ÷ F which becomes (F – 1)! = 6 which means eq.6a) (F – 1)! = 3!
Solution
eq.6a makes: (F – 1) = 3 Add 1 to both sides of the above equation: (F – 1) + 1 = 3 + 1 which makes F = 4 and makes A = B = 2×F = 2 × 4 = 8 C = 1¾×F = 1¾ × 4 = 7 D = ¾×F = ¾ × 4 = 3 E = 1½×F = 1½ × 4 = 6 and ABCDEF = 887364