Puzzle for December 25, 2020 ( )
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.
* F! is F-factorial.
Scratchpad
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Hint #1
In eq.3, replace C with E + F (from eq.2): D + E = E + F + F which becomes D + E = E + 2×F Subract E from each side of the above equation: D + E – E = E + 2×F – E which makes D = 2×F
Hint #2
eq.5 may be written as: A = (B + C + D + E + F) ÷ 5 Multiply both sides of the above equation by 5: 5×A = 5 × (B + C + D + E + F) ÷ 5 which becomes eq.5a) 5×A = B + C + D + E + F
Hint #3
In eq.1, substitute 5×A for B + C + D + E + F (from eq.5a): A + 5×A = 36 which becomes 6×A = 36 Divide both sides of the above equation by 6: 6×A ÷ 6 = 36 ÷ 6 which makes A = 6
Hint #4
In eq.4, substitute E + F for C (from eq.2), and 6 for A: E + F + E = 6 + F which becomes 2×E + F = 6 + F Subtract F from each side of the equation above: 2×E + F – F = 6 + F – F which makes 2×E = 6 Divide both sides by 2: 2×E ÷ 2 = 6 ÷ 2 which makes E = 3
Hint #5
Substitute 3 for E in eq.2: eq.2a) 3 + F = C
Hint #6
Substitute 2×F for D, and 3 for E in eq.6: F! = 2×F × 3 which becomes F! = 6×F Divide both sides of the above equation by F: F! ÷ F = 6×F ÷ F which becomes (F – 1)! = 6 which makes F – 1 = 3 Add 1 to both sides: F – 1 + 1 = 3 + 1 which makes F = 4 making D = 2×F = 2×4 = 8
Hint #7
Substitute 4 for F in eq.2a: 3 + 4 = C which makes 7 = C
Solution
Substitute 6 for A, 7 for C, 8 for D, 3 for E, and 4 for F in eq.1: 6 + B + 7 + 8 + 3 + 4 = 36 which simplifies to B + 28 = 36 Subtract 28 from both sides of the equation above: B + 28 – 28 = 36 – 28 which means B = 8 and makes ABCDEF = 687834