Puzzle for August 22, 2021 ( )
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.
* D! is D-factorial.
Scratchpad
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Hint #1
Add A to both sides of eq.3: C – A + A = D + F + A which becomes C = D + F + A Subtract D and F from both sides of the equation above: C – D – F = D + F + A – D – F which becomes eq.3a) C – D – F = A
Hint #2
Add B and D to both sides of eq.1: F – D + B + D = E – B + B + D which becomes F + B = E + D Subtract E from both sides of the equation above: F + B – E = E + D – E which becomes F + B – E = D which may be written as eq.1a) B + F – E = D
Hint #3
In eq.5, replace (C – D – F) with A (from eq.3a), and replace (B + F – E) with D (from eq.1a): eq.5a) B = A × D
Hint #4
In eq.6, replace B with A × D (from eq.5a): D! = A × D ÷ A which means D! = D The above equation makes either: D = 1 or D = 2
Hint #5
Begin checking: D = 1 ... Substituting 1 for D in eq.5a would yield: B = A × 1 which would make B = A
Hint #6
Finish checking: D = 1 ... Substituting 1 for D, and A for B in eq.2 would yield: A + 1 = A – 1 Subtracting A from both sides of the above equation would make: A + 1 – A = A – 1 – A which would make 1 = –1 However: 1 ≠ –1 which means D ≠ 1 and means D = 2
Hint #7
Substitute 2 for D in eq.5a: B = A × 2 which may be written as B = 2×A
Hint #8
Substitute 2 for D, and 2×A for B in eq.2: A + 2 = 2×A – 2 In the above equation, subtract A from both sides, and add 2 to both sides: A + 2 – A + 2 = 2×A – 2 – A + 2 which makes 4 = A and also makes B = 2×A = 2 × 4 = 8
Hint #9
Substitute 2 for D, and 8 for B in eq.1: F – 2 = E – 8 Add 2 to both sides of the above equation: F – 2 + 2 = E – 8 + 2 which becomes eq.1b) F = E – 6
Hint #10
Substitute 4 for A, 2 for D, and E – 6 for F (from eq.1b) in eq.3: C – 4 = 2 + E – 6 which becomes C – 4 = E – 4 Add 4 to both sides of the above equation: C – 4 + 4 = E – 4 + 4 which makes C = E
Hint #11
Substitute 8 for B, E for C, and E – 6 for F (from eq.1b) in eq.4: E + E – 6 = 8 + E – E which becomes 2×E – 6 = 8 Add 6 to both sides of the above equation: 2×E – 6 + 6 = 8 + 6 which makes 2×E = 14 Divide both sides by 2: 2×E ÷ 2 = 14 ÷ 2 which makes E = 7 and also makes C = E = 7
Solution
Substitute 7 for E in eq.1b: F = 7 – 6 which makes F = 1 and makes ABCDEF = 487271