Selfie Numbers - III: With Factorial and Without Square-Root - Up To Five Digits
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- Logan McBride
- 5 years ago
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1 Selfie Numbers - III: With Factorial and Without Square-Root - Up To Five Digits Inder J. Taneja 1 Abstract In previous works [11, 15, 16], the construction of Selfie numbers is done in different forms, such as in order of digits, in reverse order of digits, in increasing and decreasing orders of digits. This has been done using factorial and square-root with basic operations. This work is restricted up to five digits only with factorial and without use of square-root. Studies including square-root can be seen in author s work [22]. 1 Selfie Numbers Numbers represented by their own digits connected by certain operations are understood as Selfie numbers. These numbers are divided in two categories. Each category is again divided in two ways, i.e., one in order of digits appearing in the numbers and their reverse, and the second is in increasing and decreasing order of digits. Below are some examples of Selfie numbers. 1.1 Representations in Order of Digits and Reverse Order of Digits 24 = (2 4)!. 936 = ( 9!) 3 + 6! = (1 + 2)! 9 / = ! := 2 4! + (4! + 5) 3. Reverse Order of Digits 24 = (4!) = 6! + (3!) = 6 ( 9+2 1) = 7 + ( !)!/ = (5 ( )) Representations in Increasing and Decreasing Orders of Digits Increasing Order of Digits 936 = 3!! = (1 + 2)! = 3 7! + 8!/ := 4! + 5! = 3!! ( 4 (5! 6) 7! 8). Decreasing Order of Digits 936 = ( 9)!! = (( 9)! 6) = (8! 4)/ := ( 3 + 3!!) = 9 + (6! 5!/ 4) (3 0!). Some studied on this kind of numbers appearing in sections can be seen in Taneja [11, 15, 16]. We observe that there are number that can be represented without any order, for example, = 2 (3 5) 6! 4! = (3! + 7) 8!/ Even though these numbers are also Selfie numbers, but are not under study. 1.3 Symmetrical Representations In [15], author studied an interesting symmetrical consecutive representation of Selfie numbers, such as = 0 + 9!/ = 1 + 9!/ = 2 + 9!/ = 3 + 9!/ = 4 + 9!/ = 5 + 9!/ = 6 + 9!/ = 7 + 9!/ = 8 + 9!/ = 9 + 9!/ Formerly, Professor of Mathematics, Universidade Federal de Santa Catarina, Florianópolis, SC, Brazil. ijtaneja@gmail.com. 1
2 1.4 Unified Selfie Numbers We observe that there are numbers that can be written in all the four ways. For simplicity, we call them as unified Selfie numbers [20]. According to subsections 1.1 and 1.2, the numbers 936 and 1296 are unified Selfie numbers. See below: 936 = ( 9)! 3 + 6! = 6! + 9 3! = 3!! = ( 9)!! = (1 + 2)! 9 /6 = = (1 + 2)! 9 6 = (( 9)! 6) 2 1. More precisely, unified Selfie numbers are understood as: 1.5 Patterned Selfie Numbers Unified Selfie number = Order of digits = Reverse order of digits = Increasing order of digits = Decreasing order of digits. Madachy [4], pages discussed the idea of different kind of numbers. In [21], we called them as patterned Selfie numbers and studied extensively. See examples below: 36 = 3! = 3! = 3! = ( 4 + (2 9)!) = ( 4 + (2 9)!) = ( 4 + (2 9)!) = (( 9)! 3! 5 + 5!) = (( 9)! 3! 5 + 5!) = (( 9)! 3! 5 + 5!) 200 In [11, 15, 16], we studied extensively [ Selfie numbers having the operations, addition, subtraction, multiplication, potentiation, division, square-root and factorial i.e., +,,, ˆ, /, ],!. This work is an extension of previous work [11, 15, 16], but limited only up to 5 digits. Working with six digits with square-root and factorial there are more than possibilities. In continuation the second part - Selfie Numbers - II, [23] is on six digits without factorial. This work is restricted up to five digits and only with factorial. Studies including square-root can be seen in author s work [22]. Study on numbers in different situation can be seen in [10, 12, 13, 14, 17, 18, 19]. For some comments refer to [1, 5]. Previous work in this direction can seen in [2, 3, 6, 7, 8, 9]. 2 Selfie Numbers with Plus and Minus Signs In 1966, Madachy [4], page 167, gave examples of selfie numbers just with factorial sum: 1 = 1! 2 = 2! 145 = 1! + 4! + 5! = 4! + 0! + 5! + 8! + 5! This section deals with similar kind of numbers with factorial using only plus and minus signs. For other expressions see subsequent sections. 2.1 Digits Order 120 = ((1 + 2)! 0!)!. 144 = (1 + 4)! + 4!. 145 = 1 + 4! + 5!. 660 = 6! = (7 1)! = 7 + 3!! + 3! = ( 1 + 4)!! + 3!! = ( 1 + 4)!! + (4 0!)!! = 1 + 4! + 6! + 3!! = ( 1 + 4)!! + 6! + 4! = 4! + ( 2 + 9)! 6! = 4 3!! ! = (4 + 3)! (2 + 0!)!! = (5 0!)! + (1 + 6)! = (5 0!)! ! = 5 0! + (3 + 4)! = (5 0! + 3)! = 5! + (1 + 6)! = 5! + (1 + 6)! = 5! + (1 + 6)! = 5! + (1 + 6)! = 5! + (1 + 6)! = 5! + (1 + 6)! = 5! + (1 + 6)! = 5! + (1 + 6)! = 5! + (1 + 6)! = 5! + (1 + 6)! = 5! ! = 5! + ( 1 + 8)! + 4!. 2
3 5637 = 5! + 6! 3 + 7! = 6! 4 + 7! + 6! = 1 0! 0! + 7! + 7! = (3 + 5)! 2 7! = (3 + 5)! + 2 7! = 3! (5 + 2)! + 8! = (3 + 5)! (3! + 0!)! + 4! = 3!! 5! + 8! 7! = 3!! 5! + 8! (8 0!)! = 3!! ! = 3!! + 8! 7 5! 3!! = 3!! + 8! = 3!! + 8! + 8 6! 4! = 3!! + 8! 8 6! = 3! ! = 3!! (9 4)! + 8! = 3!! (9 4)! + 8! = 3!! (9 4)! + 8! = 3!! (9 4)! + 8! = 3!! (9 4)! + 8! = 3!! (9 4)! + 8! = 3!! (9 4)! + 8! = 3!! (9 4)! + 8! = 3!! (9 4)! + 8! Reverse Order of Digits 144 = 4! + (4 + 1)!. 145 = 5! + 4! = 5 + ( 1 + 7)!. 733 = 3! + 3!! = 5 + 3!! + (4 1)!! = 3!! + 6! + 4! = 4! + 6! + (4 1)!! = 6! + (9 2)! 4! = 7! + 1 3!! = 7! 59 4! = 7! = (6 + 1)! ( 0! + 5)! = 7! + 1 ( 0! + 5)! = (4 + 3)! 0! = ( !)! = 0 + (6 + 1)! + 5! = 1 + (6 + 1)! + 5! = 2 + (6 + 1)! + 5! = 3 + (6 + 1)! + 5! = 4 + (6 + 1)! + 5! = 5 + (6 + 1)! + 5! = 6 + (6 + 1)! + 5! = 7 + (6 + 1)! + 5! = 8 + (6 + 1)! + 5! = 9 + (6 + 1)! + 5! = 5! + 7! = 4! + (8 1)! + 5! = 7! 3 + 6! 5! = 3!! (9 4)! + 8! = 3!! ! = ! 3!! = ( 3+9)!+(6+2)!+4! = 4! 0! + (1 + 7)! 5! = (4+0!)!+( 1+9)! = ! = 4! (0! + 2)! + 8! = 4! ! = 4! ! = 4! (0!+2)!+(9 0!)! = 4 0! 3!+( 0!+9)! = (4 0!)!+(3+1+4)! = 4 + ( )! = (40 32)! = (40 32)! = (40 32)! = (40 32)! = (40 32)! = (40 32)! = (40 32)! = (40 32)! = (40 32)! = (40 32)! = 40 + (3 + 5)! = 6! + 7! 4 + 6! = 7! + 7! 0! 0! = 4 7! 2 + (5 + 3)! = 6 7! (5 + 3)! = 4! (0! + 3!)! + (5 + 3)! = 5 7! + 8! 5! + 3!! = 3!! 5! 7 + 8! 3!! = 4! 6! ! 3!! = 6! 6! 8 + 8! 3! = 1 + 8! ( 4 + 9)! 3!! = 2 + 8! ( 4 + 9)! 3!! = 3 + 8! ( 4 + 9)! 3!! = 4 + 8! ( 4 + 9)! 3!! = 5! + 8! !! = 6!+8! ( 4+9)!+3! = 7 + 8! ( 4 + 9)! 3!! = 8 + 8! ( 4 + 9)! 3!! = 9 + 8! ( 4 + 9)! 3!! = 8! 3!! = 4! + (2 + 6)! (9 3)! = 5! + (7 + 1)! 0! 4! = 5! + (9 1)! 0! = 5 + 8! (2 + 0!)! 4! = 7 + 8! 2 04! = 8! = 9 + 8! ! = (9 0!)! 3! 0! = 3 + (1 + 3! + 0!)! = ! = 40 + ( !)! = (4 + 0!)! + 4 3! + 8! = (4 + 04)! + (4 + 0!)! = (4 + 0!)! + (4 + 4)! = (4 + 0!)! + (4 + 4)! = (4 + 0!)! + (4 + 4)! = (4 + 0!)! + (4 + 4)! = (4 + 0!)! + (4 + 4)! = (4 + 0!)! + (4 + 4)! = (4 + 0!)! + (4 + 4)! = (4 + 0!)! + (4 + 4)! = (4 + 0!)! + (4 + 4)! = (4 0!)!+4!+5!+8! = (4 + 0!)! + 5! + 8! + 4! = 4! + 0! + 5! + 8! + 5! = (4 1)!! + 0! 3 + 8! = (4 + 4)! 6! 3 + 7! = 4! + (5 + 3)! + 7! = 8! 0! 5! 1 + 8! = 8! 0! 5! + ( 1 + 9)! = 8! ! + 8! = 8! + (0! + 7)! + 5! = 8! + (0! + 7)! + (6 0!)! = (4 1)!+(3+0!+4)! = 5 + (13 0! 4)! = ( )! = 8! = 8! 3! (0! + 4)! = 0 + (4 + 4)! + (0! + 4)! = 1 + (4 + 4)! + (0! + 4)! = 2 + (4 + 4)! + (0! + 4)! = 3 + (4 + 4)! + (0! + 4)! = 4 + (4 + 4)! + (0! + 4)! = 5 + (4 + 4)! + (0! + 4)! = 6 + (4 + 4)! + (0! + 4)! = 7 + (4 + 4)! + (0! + 4)! = 8 + (4 + 4)! + (0! + 4)! = 9 + (4 + 4)! + (0! + 4)! = 8! + 5! (4 0!)! + 4! = 4! + 8! + 5! + (0! + 4)! = 5! + 8! + 5! + 0! + 4! = 6! + (3! + 0! + 1)! = 8! + 3!! + 0! = 7! 3 6! + (4 + 4)! = 7! 7 + (3 + 5)! + 4! = 4! + 8! + (3! + 5 4)! = (8! 1 (5! + 0!)) + 8! = (9 1)! 5! 0! + 8! = 5 + 8! ! = (5 + 3)! 6 + 0! + 8! = 5! 5 + (7 + 0!)! + 8!. 3
4 2.3 Increasing Order of Digits 120 = ( 0! + (1 + 2)!)!. 144 = (1 + 4)! + 4!. 145 = (1 + 4!) + 5!. 720 = (0! 2 + 7)!. 733 = 3!! + 3! = 4! + ( 4 + 7)!! = 1 + 3!! + 4! + 6! = ( 1 + 4)!! + 4! + 6! = 1 3!! 4 + 7! = (0! + 2)!! + (3 + 4)! = 0! ( 1 + 5)! + 7! = 0! + (3 + 4)! = ( 0! )! = 0! ! = 0! + ( )! = ( 0! 0! )! = 1 + 5! ! = ! + 7! = ! + 7! = ! + 7! = ! + 7! = ! + 7! = 3 5! + 6! + 7! = 4 + 6! + 6! + 7! = 0! 0! 1 + 7! + 7! = 3!! 3!! 3 7! + 8! = (1+3!)!+4! 5!+8! = 2 + (3 + 5)! 6 7! = (0! )! + 8!. 2.4 Decreasing Order of Digits 120 = ((2 + 1)! 0!)!. 144 = 4! + (4 + 1)!. 145 = 5! + 4! = 6! = 7 + 3! + 3!!. 744 = (7 4)!! + 4! = 6! + 4! + 3!! = 6! + 4! + (4 1)!! = 6! 4 + (3! + 1)! = 7! 4 3!! = (4 + 3)! (2 + 0!)!! = 9 + 7! = 7! (5 1)! + 0! = 5 + (4 + 3)! 0! = 5 + ( !)! = 7! ! = ( )! 0! = 7! ! = 7! + 5! = 7! + 5! = 7! + 6! 5! = 3!! 3!! 5! 7 + 8! = 3!! 4! 6! ! = 3!! 6! ! = 3!! 4 5! + 8! = 3!! + (3 + 5)! = 3!! + ( ! 9) = (0!+0!+3!)! ( 6+9)!! = 0! ( 3+6)!!+8! = 0! 14 5! + 8! = (0! + 2)! ! = 0! ! = 02 4! 7 + 8! = ! = 02 4! + 8! = (0! !)! = ( 0! )! = (0! !)! = 0! + ( )! = 0! + ( )! = ( 0! )! = ( 0! )! = ( 0! )! = ( 0! )! = ( 0! )! = ( 0! )! = (0! + 0! + 3!)! + 3! = ! + 8! = (0! + 0! + 3!)! 4 + 4! = 0! (4 + 4)! = 0!+(3!+3! 4)!+4! = 0! ! = 7! + 6! + 6! = 7! + 7! 1 0! 0! = 8! 7! 3 3!! 3!! = 8! 5! + 4! (3! + 1)! = 7! 6 + (5 + 3)! = 8! ( !)! = 8! 7! 5 5! + 3!! = 8! 7 5! 3!! 3!! = 8! 8 6 6! 3!! = 9 + 8! 5! 4 3!! = 9 + 8! 53 3!! = (5 + 3)! 3!! = ! + 5 3!! = (9 6)!!+(3!+0!+0!)! = 9 + 8! (6 3)!! 0! = (9 6)!!+4!+(3!+2)! = 9 + 8! = 8! 5! 41 0! = 8! ! + 0! = 8! ! = 8! = (4 + 4)! ( !)! = 0! ! = ( 0!+3!)!+(4+4)! = 0! + (1 + 4)! + (4 + 4)! = (0! + 4)! (4 + 4)! = 0! (4 + 4)! + 5! = (0! + 4)! + (4 + 4)! = (0! + 4)! + (4 + 4)! = (0! + 4)! ! = (0! + 4)! + (4 + 4)! = ( 0!+4)!+4!+5!+8! = (0! + 4)! ! = (0! + 4)! ! = (0! + 4)! ! = (0! + 4)! ! = (0! + 4)! ! = (0! + 4)! + 4! + 5! + 8! = 0! + 4! + 5! + 5! + 8! = 0! ! = (0! !)! 4 + 6! = 0! !! 4 + 8! = (0! + 0! + 1)!! + (4 + 4)! = 3 + (4 + 4)! 6! + 7! = 2 + (3 + 4)! 5! + 8! = (4 + 4)! + 5! 5 + 7! = 0! 1 5! + 8! + 8! = ! + 8! = 0! + (3 + 5)! 6 + 8! = 0! ! + 8! = 0! + (14 6)! + 8! = ( 0! + 5)! 6 + 8! + 8! = (0! !)! + 6! + 8! = ( )! = 4 + 3! + (3! + 0! + 0!)! = 4! 4 + (3! + 0! + 0!)! = 5 + (4 + 4)! = 8! ! = 8! ! = 8! ! 0! = 5+(4+4)!+(3! 0!)! = (4 + 4)! + (4 + 1)! + 0! = 4 + (4 + 4)! + (4 + 0!)! = 5 + (4 + 4)! + (4 + 0!)! = 6 + (4 + 4)! + (4 + 0!)! = 7 + (4 + 4)! + (4 + 0!)! = 8 + (4 + 4)! + (4 + 0!)! = 9 + (4 + 4)! + (4 + 0!)! = 8! + 5! + 4! (4 0!)! = 8! + 5! + 4! + (4 + 0!)! = 8! = 6! 4 + (3! !)! = 8! 4 + 3!! ! = (4 + 4)! + (1 + 0! + 0!)!! = (1 + 3!)! 4 6! = 7! 6! + (4 + 4)! 3. 4
5 45238 = 8! 5! + (4 + 3)! = 7! + 5! 5 + (4 + 4)!. 3 Unified Selfie Numbers = 8! + 8! 5! 1 0! = 8! 6 + (5 + 3)! + 0! = 8! + 8! 6 + (5 0!)! = 8! + 6! + (3! !)!. According to subsection 1.4, below are examples of unified Selfie numbers. The study is limited only up to 6 digits without factorial. It uses only square-root with other basic operations: 36 = 3! 6 = 6 3!. 120 = ((1 + 2)! 0!)! = ( 0! + (2 + 1)!)! = ( 0! + (1 + 2)!)! = ((2 + 1)! 0!)!. 143 = 1 + 4! 3! = 3! 4! 1 = 1 + 3! 4! = 4! 3! = (1 + 4)! + 4! = 4! + (4 + 1)!. 145 = 1 + 4! + 5! = 5! + 4! = 3 5! 5 = 5 + 5! = 4 (5! 6) = ( 6 + 5!) = !! = 3!! 1 7 = 1 3!! 7 = 7 + 3!! = (7 (2 0)!)! = (0! 2 + 7)!. 733 = 7 + 3!! + 3! = 3! + 3!! + 7. k744 = (7 + 4!) 4! = 4! (4! + 7) = 1 ( 4 + 3!!) 2 = 2 (3!! 4) 1 = 1 2 (3!! 4) = ( 4 + 3!!) = !! + 6! = 6! + 3!! 4 1 = 1 3!! 4 + 6! = 6! 4 + 3!! = ( 1 + 4)!! + (4 0!)!! = ( 0! + 4)!! + (4 1)!! = (0! + 1) (4!/4)! = (4!/4)! (1 + 0!). k1463 = 1 + 4! + 6! + 3!! = 3!! + 6! + 4! 1 = 1 + 3!! + 4! + 6! = 6! + 4! + 3!! 1. k1464 = ( 1 + 4)!! + 6! + 4! = 4! + 6! + (4 1)!! = ( 1 + 4)!! + 4! + 6! = 6! + 4! + (4 1)!! = (2 + 1) 6! + 0 = 0 + 6! (1 + 2) = 0 + (1 + 2) 6! = 6! (2 + 1) = (2 + 1) 6! + 1 = 1 + 6! (1 + 2) = 1 + (1 + 2) 6! = 6! (2 + 1) = (2 + 1) 6! + 3 = 3 + 6! (1 + 2) = (1 + 2) + 3 6! = 6! = (2 + 1) 6! + 6 = 6 + 6! (1 + 2) = (1 + 2) 6! + 6 = 6 + 6! (2 + 1) = (2 + 1) 6! + 7 = 7 + 6! (1 + 2) = (1 + 2) 6! + 7 = 7 + 6! (2 + 1) = (2 + 1) 6! + 8 = 8 + 6! (1 + 2) = (1 + 2) 6! + 8 = 8 + 6! (2 + 1). 169 = (2 + 1) 6! + 9 = 9 + 6! (1 + 2) = (1 + 2) 6! + 9 = 9 + 6! (2 + 1) = (5 + 2)!/2 + 0 = 0 + (2 + 5)!/2 = ( 0! + 22) 5! = (2 + 5)!/ = ( ) 5 = = 1 (2 + 3) 5 = = 3! 4! 4! 8 = 8 + 4! 4! 3! = (3!! 4! 5) 5 = 5 ( 5 4! + 3!!) = 3! 4! + 5 6! = 6! 5 4! 3! = 3!! = !! = 3!! = !! = 3!! 5 9 0! = 0! !! = 0! + 3!! 5 9 = !! 0! = 3!! = !! = 1 3!! 5 9 = !! 1. 5
6 3599 = 3!! 5 ( 9 + 9)! = ( 9 + 9)! + 5 3!! = 3!! 5 9/9 = 9/ !! = 3! 600 = 0 + ( 0! + 6) 3!! = ( 0! + 03)! 6! = 6! (3! 0! 0!) = 3!! (6 0!) + 1 = 1 + ( 0! + 6) 3!! = 0! + ( 1 + 3!) 6! = 6! (3! 1) + 0! = 3!! (6 0!) + 4 = 4 + ( 0! + 6) 3!! = (0! + 3!!) 4 + 6! = 6! + 4 (3!! + 0!) = 3!! (6 0!) + 5 = 5 + ( 0! + 6) 3!! = 0! + 3! + 5 6! = 6! 5 + 3! 0! = 3!! (6 0!) + 6 = 6 + ( 0! + 6) 3!! = ( 0! + 3!) 6! + 6 = 6 + 6! (3! 0!) = 3!! (6 0!) + 7 = 7 + ( 0! + 6) 3!! = ( 0! + 3!) 6! + 7 = 7 + 6! (3! 0!) = 3!! (6 0!) + 8 = 8 + ( 0! + 6) 3!! = ( 0! + 3!) 6! + 8 = 8 + 6! (3! 0!) = 3!! (6 0!) + 9 = 9 + ( 0! + 6) 3!! = ( 0! + 3!) 6! + 9 = 9 + 6! (3! 0!) = (3 + 6! + 2) 5 = 5 (2 + 6! + 3) = (2 + 3) (5 + 6!) = (6! + 5) (3 + 2) = (3! + 6!) (3! 0!) = ( 0! + 3!) (6! + 3!) = ( 0! + 3!) (3! + 6!) = (6 + 3!!) (3! 0!) = 3! (6 + 3!!) 6! = 6 (3! + 6!) 3!! = 3!! + 3! (6 + 6!) = (6 + 6!) 3! 3!! = 4! (3!! 1)/4 = 4! ( 1 + 3!!)/4 = ( 1 + 3!!) 4!/4 = 4!/4 (3!! 1) = 4 + 3! 1 6! = 6 1 3!! 4 = (1 + 3!)! 4 6! = 6! 4 + (3! + 1)! = 4 3!! ! = 7! + 1 3!! 4 = 1 3!! 4 + 7! = 7! 4 3!! = (4 + 3)! (2 + 0!)!! = (0! + 2)!! + (3 + 4)! = 4! + 3! (3!! 2) = ( 2 + 3!!) 3! + 4! = (4 + 3!!) 3! 7 = 7 + 3! (3!! + 4) = 3! 3!! + 4! 7 = 7 + 4! + 3!! 3! = 4! (3!! + 4)/4 = 4! (4 + 3!!)/4 = 3! (4 + (4!/4)!) = (4 + (4!/4)!) 3! = 4! (4! + 6!)/4 = 4! (6! + 4!)/4 = (4! + (4!/4)!) 6 = 6 (4! + (4!/4)!) = (5 0!)! ! = 7! + 1 ( 0! + 5)! = 0! ( 1 + 5)! + 7! = 7! (5 1)! + 0! = 5 0! + (3 + 4)! = (4 + 3)! 0! 5 = 0! + (3 + 4)! 5 = 5 + (4 + 3)! 0! = (5 0! + 3)! 5 = ( !)! 5 = ( 0! )! 5 = 5 + ( !)! = ! = 7! = 0! ! = 7! ! = 5! ! = 7! ! + 1 = ! + 7! = 7! + 5! = ( 5!/5 + 6!) 8 = 8 (6! 5!/5) = 5! + 6! 3 + 7! = 7! 3 + 6! 5! = 3 5! + 7! + 6! = 7! + 6! 5! = 5 + 7! + 6 5! = 5! 6 + 7! + 5 = 5 + 5! 6 + 7! = 7! + 6 5! = ((6 3)!! 9) 9 = 9 ( 9 + ( 3 + 6)!!) = (( 3 + 6)!! 9) 9 = 9 ( 9 + (6 3)!!) = 6! 4 + 7! + 6! = 6! + 7! 4 + 6! = 4 + 6! + 6! + 7! = 7! + 6! + 6! = 8 + 6! 3! 2 = 2 3! 6! = 1 + (0! + 0!) ( 6 + 7!) = (7! 6) (0! + 0!) 1 = 0! + (0! + 1) ( 6 + 7!) = (7! 6) (1 + 0!) 0!. 6
7 10073 = 1 + (0! + 0!) (7! 3) = ( 3 + 7!) (0! + 0!) 1 = 0! + (0! + 1) ( 3 + 7!) = (7! 3) (1 + 0!) 0! = (1 + 0!) (0! + 7! 4) = ( 4 + 7! + 0!) (0! + 1) = (0! + 0!) ( !) = (7! 4 + 1) (0! + 0!) = 1 0! 0! + 7! + 7! = 7! + 7! 0! 0! 1 = 0! 0! 1 + 7! + 7! = 7! + 7! 1 0! 0! = (1 + 0!) ( 0! + 8)! + 0 = 0 + (8 0!)! (0! + 1) = (0! + 0!) ( )! = (8 1 0!)! (0! + 0!) = (1 + 0!) ( 0! + 8)! + 1 = 1 + (8 0!)! (0! + 1) = 0! + (0! + 1) ( 1 + 8)! = (8 1)! (1 + 0!) + 0! = (1 + 0!) ( 0! + 8)! + 7 = 7 + (8 0!)! (0! + 1) = 0! + (0! + 1) 7! + 8 = 8 + 7! (1 + 0!) 0! = (1 + 0!) ( 0! + 8)! + 8 = 8 + (8 0!)! (0! + 1) = (0! + 0!) ( 1 + 8)! + 8 = 8 + (8 1)! (0! + 0!) = (1 + 0!) ( 0! + 8)! + 9 = 9 + (8 0!)! (0! + 1) = (0! + 0!) ( 1 + 8)! + 9 = 9 + (8 1)! (0! + 0!) = 1 + (0! + 0!) (9 + 7!) = (7! + 9) (0! + 0!) 1 = 0! + (0! + 1) (7! + 9) = (9 + 7!) (1 + 0!) 0! = (10! 7!)/(8!/5!) = 5!/8 ((7 0!)! 1) = ( 0! + (1 + 5)!) (7 + 8) = (8 + 7) ((5 + 1)! 0!) = ( !!) 4 4 = 4 4 (3!! 11) = ((1 + 2)!! + 2) ( 7 + 4!) = (4! 7) (2 + (2 + 1)!!) = ((1 + 2)!! + 2) (4! 7) = ( 7 + 4!) (2! + (2 + 1)!!) = (1 + 2)! = (8 8) 2 (2 + 1) = ( 2 + 3!!) = (3!! 2) (9 2) 1 = ( 2 + 3!!) 9 = 9 (3!! 2) = (1 + 2) ( 9 + 3! 3!!) = (3! 3!! 9) (2 + 1) = ((1 + 2)!! 3 + 3!!) 9 = 9 (3!! 3 + (2 + 1)!!) = ! + 0 = 0 + 6! = ! 9 = 9!/( !) = ! 1 = 1 + 6! = ( 1 + 1)! + 2 6! 9 = 9 6! 2 + ( 1 + 1)! = ! 2 = 2 + 6! = ! 9 = 9 6! = ! + 3 = 3 + 6! = (3!! + 6!) 9 = 9 (6! + 3!!) = ! + 6 = 6 + 6! = (1 + 2)! + (6! + 6!) 9 = 9 (6! + 6!) + (2 + 1)! = ! + 9 = 9 + 6! = 1 2 6! = ! = ( 1 3! + 5!) 5! 5 = 5 + 5! (5! 3! 1) = ! (5! 7) = ( 7 + 5!) 5! = ( 1 + 3!)! (5! 6 0!) = ( 0! 6 + 5!) (3! 1)! = ( 0! + ( 1 + 3!)!) 5! 6! = 6! + 5! ((3! 1)! 0!) = ((1 + 3)! 5) (6! 6) = ( 6 + 6!) ( 5 + (3 + 1)!) = 1 (3 8) 2 4! = ( ) 3 1 = (1 + 23) 4!/8 = (8 4)! 3 (2 1) = ( ( 3 + 3!!)) 5 = 5 ((3!! 3) 4 1) = ( !! 4) 5 = 5 (4 (3!! 3) 1) = (1 + 4)! (5! + 1) 5 = 5! (1 + 5!) 4 1 = (1 + (1 + 4)!) 5! 5 = 5 + 5! ((4 + 1)! + 1) = 1 + 4! ( 5! + 6! + 7) = (7 + 6! 5!) 4! = (1 + 5)! (1 + 20) = (0! + 2) ( )! = (0! ) (2 + 5)! = (5 + 2)! ( !) = 1 (5 5 8) 5 = 5 ( ) 1 = 1 5 (5 5 8) = = 1 5 (6+2 5)! = = 1 (25/5) 6 = (6 5) 5 2+1! = ! 1 = ( 1 + 3!) = ( )! = (3 1+1)! = ! + 5 = 5 3! = ( 1 + 3) = ! 1. 7
8 15655 = 1 5 ( ) = ( ) 5 1 = (1 + 5) = = 16 3 (8 4) = (8/(6 4)) 3!+1 = ( ) 6+8 = ((4 + 8)/3) = 1 6! + ( 4 + 6!) 4! = 4! (6! 4) 6! 1 = 1 4! ( 4 + 6!) 6! = 6! + (6! 4) 4! = ( 1 + 6!) (5 3! 7) = ( 7 + 3! 5) (6! 1) = ( 1 + 3!!) (5 6 7) = ( ) (3!! 1) = ( 1 + 6! 9 4) 4! = 4! ( ! 1) = (1 + 4)! + 4! (6! 9) = ( 9 + 6!) 4! (4 + 1)! = 1 + (6! 9) 4! 5! = 5! + 4! ( 9 + 6!) + 1 = 1 4! (5 6! + 9) = ( 9 + 6! 5) 4! = ((1 + 7)/2)! (3!! 2) = ( 2 + 3!)! ( 2 + (7 1)!) = ((1 + 2)!! 2) ( 3 + 7)! = (7 3)! ( 2 + (2 + 1)!!) = 1 + (7 3)! (0! + 3!!) = (3!! + 0!) ( 3 + 7)! 1 = 0! + (1 + 3!!) ( 3 + 7)! = (7 3)! (3!! + 1) 0! = (( 1 + 7)! + 3) 4! 6 = 6 + (4! (3 + (7 1)!)) = (1 + 3!!) 4! = ! (3!! + 1) = 1 9 (3 6! 8) = ( 8 + 6! 3) 9 1 = (1 3 6! 8) 9 = 9 ( 8 + 6! 3 1) = (((2 + 0!)! + 1)! 4) 4 = 4 ( 4 + (1 + (0! + 2)!)!) = ((0! + (1 + 2)!)! 4) 4 = 4 ( 4 + ((2 + 1)! + 0!)!) = (2 + 1)! ( 4! + 5 6!) = (6! 5 4!) (1 + 2)! = ! 6 = (6! 5 4!) (2 + 1)! = 2 ( 2 + 3! 3! /4) = 4 + 3! 3 2 /2 = (2 3) 3! /2 4 = 4 + (2 3) 3! / = (2 3 3 ) 2 8 = (8 2) 3+3 /2 = ! 8 = 8 (3 3 2) = 2 (3 + 3! 3! /4) = ( ! 3! )/ = (2 3!! + 4!) 2 4 = 4 2 (4! + 3!! 2) = (2 2)! (3!! ) = ( !!) (2 + 2)! = ( 2 + ( 3 + 6)!!) (9 + 4!) = (4! + 9) ((6 3)!! 2) = (2 3!!) ( 4 6 9) = ( ) (3!! 2) = 2 3!! + 7! 5 5 = ! 3!! 2 = 2 3!! ! = 7! 5 5 3!! = ((1 + 6)! 7) = (7! 6 1) = ( 6 + 7!) = (7! 6) = (1 + 7! 3!) = ( 3! + 7! + 1) 5 2 = (1 + 2) ! = 7! = 4! + 7! = ! 4! = 1 2 4! + 5 7! = 7! 5 4! = 25 ( 1 + 7!/5) = 5 7! = ! = (7! 5) 5 (2 1) = (2 + 5)! ((2 + 0!)! 0!) = ( 0! + (0! + 2)!) (5 + 2)! = ( 0! + (0! + 2)!) (2 + 5)! = 5 (( )! + 0!)! = ((2 + 5)! + 2 0!) 5 = 5 ( 0! (5 + 2)!) = ((0 2)! + (2 + 5)!) 5 = 5 ((5 + 2)! + (2 0)!) = ((2 + 5)! ) 5 = 5 ( (5 + 2)!) = ( (2 + 5)!) 5 = 5 ((5 + 2)! ) = (7! + 5! 8) = ( 8 + 5! + 7!) 5 2 = (5! + 7! 8) = ( 8 + 7! + 5!) = 3! (0! 2 + (3 + 4)!) = ((4 + 3)! (2 0)!) 3! = (0! + 2)! + 3! (3 + 4)! = (4 + 3)! 3! (2 + 0!)! = 3! (0! + 2 3)! 5 = (0! + 3!)! = (0! + 2 3)! 3! 5 = 5 + 3! ( !)! = 7! = ! = ! 7! = 7! 3! = 3! (0! )! + 3 = (0 + 3!!) = (0! + 2) + 3! (3 + 4)! = (4 + 3)! 3! !. 8
9 30264 = 3! (((0/2)! + 6)! + 4) = (4 + (6 + (2 0)!)!) 3! = ((0! + 2 3)! + 4) 6 = 6 (4 + ( !)!) = 3 0! ! = 7! 6 + (2 + 0!) 3 = (0! + 2) ! = 7! ! = 3! ((0! + 2)! + 7!) 3 = ((3! + 7!) 2 0!) 3 = 0! 2 + 3! (3! + 7!) = 7! 3! ! = (6 + 7!) 2 (0 + 3) = 3 (0 + 2) (7! + 6) = 02 3 (6 + 7!) = (7! + 6) 3 2 0! = 3 (0! + 2 (9 + 7!)) = ((7! + 9) 2 + 0!) 3 = 0! ! (7! + 9) = (9 + 7!) 3! ! = 3! (0! + 3!)! + 5! 5 = 5! 5 + 3! (0! + 3!)! = (0! + 3!)! 3! 5 + 5! = 5! 5 + 3! (3! + 0!)! = (3! + 0!) (3 + 6!) 6 = 6 (6! + 3) (0! + 3!) = (0! + 3!) (3 + 6!) 6 = 6 (6! + 3) (3! + 0!) = (3!! + 1) (6 + 6) + 8! = 8! (6 + 6) (1 + 3!!) = (1 + 3!!) (6 + 6) + 8! = 8! (6 + 6) (3!! + 1) = 3!! (2 + 3) 9 5 = 5 9 3!! 2 3 = !! 5 9 = 9 5 3!! = 3! (2 + 7!/4) + 8! = 8! (4 + 7!/2) 3 = 2 ( 3! ) 8 = (8 + 7)!/(5 + 3!)! = ( ) 6 /8 = ((8 6) 7 2 ) 3 = 2 3! = (8 (7 6)) = 3! (3!! 4!) (0 + 8) = 8 (0 4! + 3!!) 3! = ( (0! + 3)! + 3!!) 48 = 8 ( 4! + 3!!) 3! 0! = (3!! 3!) (55 8) = ( ) ( 3! + 3!!) = 3!! + 3! (6! 4) 8 = 8 ( 4 + 6!) 3! 3!! = 3!! + (3!! 4) 6 8 = 8 6 ( 4 + 3!!) 3!! = 3!! 3! 8 3!! 5 = 5 + 3!! 8 3! 3!! = 3 3 3!! 5 + 8! = 8! !! = 3 3!! + 8! 3!/7 = (7! 3!!) 8 3!! 3 = 3!! 3!! 3 7! + 8! = 8! 7! 3 3!! 3!! = 3!! + 3! (8 6! + 4) = (4 + 6! 8) 3! 3!! = 3!! + 3! (4 + 6! 8) = (8 6! + 4) 3! 3!! = 3! + 4! 2 (6! 6) = (6! 6) 2 4! 3! = 2 ( 3 + 4! ( 6 + 6!)) = (( 6 + 6!) 4! 3) 2! = 3! ( 4!/3! + 6!) 8 = 8 ((6 3)!! 4) 3! = ((3 + 3)! 4) 6 8 = 8 6 ( 4 + (3 + 3)!) = (3 (4 3!!) 8) 6 = 6 (8 (3!! 4) + 3) = 3! (3 + ( 4 + 6!) 8) = (8 (6! 4) + 3) 3! = 3!! 4! + ( 4 + 6!) 4! = ( 4 + 6!) 4! + 4! 3!! = 3!! 4! + 4! ( 4 + 6!) = (6! 4) 4! + 4! 3!! = 3!! (4! + 4!) 9 7 = (4! + 4!) 3!! = 3!! (4! + 4!) 7 9 = (4! + 4!) 3!! = (3!! 4! 5 4) 2 = 2 ( ! 3!!) = 2 (3!! 4! 4 5) = ( ! 3!!) = 3 (4 5! 4! 5) = ( ! 4!) 3 = 3 (4! 4 5! 5) = ( 5 + 5! 4 4!) = 3!! (4! + 5!/5) 5 = 5 + (5!/5 + 4!) 3!! = 3 + 4! 5! (5 + 7) = (7 + 5) 5! 4! = 3! (4 6! + 3!) 2 = 2 (36 + 4! 3!!) = (2 3!! + 3) 4 6 = 6 4 (3 + 3!! 2) = (8 6! 6 + 4!) 3! = 3! (4! 6 + 6! 8) = (3! (4 + 6!) 8) 8 = 8 ( 8 + (6! + 4) 3!) = ( 3 4! + 7!) 7 3 = (7! 4! 3) = 3 + ( 3 4! + 7!) 7 = 7 (7! 4! 3) = 3!! + (4! !) 7 = 7 (7! !) 3!! = ( 3! 5 + 0! + 7!) 7 = 7 (7! + 0! 5 3!) = ( (0! + 3)! 5 + 7!) 7 = 7 (7! 5 (3 + 0!)!) = 3! (5! 2 + 6! 8) = (8 6! 2 + 5!) 3! = ( 2 3!! 5!) 6 + 8! = 8! 6 (5! + 3!! + 2) = ( ) 7! 5 = (2 5 3)! = (2 + 3!!/5!)! 5 7! = 7! 5 + (5 + 3!/2)!. 9
10 35276 = (3 + 5)! + 2 7! 6 = 6 7! (5 + 3)! = 2 + (3 + 5)! 6 7! = 7! 6 + (5 + 3)! = 3! ( 2 + 7!) 7 = 7 (7! 2) ! = ! = 7 (7! 5) = 3! + (5 + 2) 7! 8 = 8 + 7! (2 + 5) + 3! = 2 + 3!! (57 8) = 8! + 7! = 3! + (5 + 4! + 7!) 7 = 7 (7! + 4! + 5) 3! = 3! + (4! !) 7 = 7 (7! !) 3! = ( 3!! + 5! 4) 8 + 8! = 8! + 8 ( 4 + 5! 3!!) = ( 3!! 4 + 5!) 8 + 8! = 8! + 8 (5! 4 3!!) = 3!! 5! + 8! 7! 5 = 5 7! + 8! 5! + 3!! = 35 (5! + 7) + 8! = 8! 7! 5 5! + 3!! = (3 + (4! + 6!) 7) 7 = 7 (7 (6! + 4!) + 3) = (3 + (6! + 4!) 7) 7 = 7 (7 (4! + 6!) + 3) = 3!! 6 (6! + 7) + 8! = 8! (7 + 6!) 6 + 3!! = 3!! (6! 7) 4 + 8! = 8! 4 ( 7 + 6!) 3!! = 3!! 4 (6! 7) + 8! = 8! + (7 6!) 4 3!! = 3 (6! 7) 5 + 8! = 8! 5 ( 7 + 6!) + 3 = 3 5 (6! 7) + 8! = 8! + (7 6!) = 3 6! + 8! 6 4 = (4 6 8) 6/3 = 3! (4! 6 6!) + 8! = 8! (6 + 6) 4 /3! = 3! + 7! 4 + 3!! 4! = 4! 3!! + 4 7! 3! = 3! + 3!! (4! + 4 7) = ( !) 3!! 3! = 3!! ( !) + 4 = 4 + (4! + 4 7) 3!! = 3!! 4! ! = 7! ! 3!! = (3! + 7) 4 6! + 4! = 4! + 6! 4 (7 + 3!) = 3!! 4! + 4 (6 + 7!) = (7! + 6) 4 + 4! 3!! = 3 + 7! = 5 (0! + (8! + 7!)/3!) = ! = 8 7! + 5 3!+0! = (3! + 8 6) (6! 4) = ( 4 + 6!) ( !) = 3! ! = ( ) ( 4 + 3!!) = (3 + 8)!/7! + 3! 6 = 6! + 3! 7 8! 3! = ( !) 8 = 8 (7! 6 33) = 3!! + 8! 7 5! 3!! = 3!! 5! 7 + 8! 3!! = 3!! 3!! 5! 7 + 8! = 8! 7 5! 3!! 3!! = 3!! + 8! 7 5! 5 = 5 5! 7 + 8! 3!! = 3!! 5 5! 7 + 8! = 8! 7 5! 5 3!! = ! + 9!/9 = (7! 8 3!!) = ( 3!! + 7! 8) 9 9 = ( 8 + 7! 3!!) = 2 3!! + 8! 8 3! = 3! 8 + 8! 3!! 2 = 2 (3!! + 3 8) + 8! = 8! ( !!) = 3!! + 8! + 8 6! 4! = 4! 6! ! 3!! = 3!! 4! 6! ! = 8! + (8 + 6!) (4 3!) = 3!! + 8! 8 6! 6 = 6! 6! 8 + 8! 3! = 3!! 6! ! = 8! 8 6 6! 3!! = ( 3!! + (8 + 8!)/8) 9 = 9 ((8 + 8!)/8 3!!) = (3! 9 3 3) 9 = 9 (3 3! 9 3 ) = ( 3 + 3! (3!! + 9)) 9 = 9 (9 3 3! 3) = 3 6 3! 9 3 = 3 9 /3 6 3 = 3 + 3! = ! = 3 9 (3 6/6) = 6 (6 3) 9 /3 = 3! (3 6) 6 9 = 9 6 (6 3) 3! = = ( )/3 = 3 + 3! (6! + 9) 9 = 9 (9 + 6!) 3! = = ( )/3 = 3 + 3! (6! + 9) 9 = 9 (9 + 6!) 3! = (3!! 9 4! + 5!) 6 = 6 (5! 4! + 9 3!!) = 3! ( 4! + 5! + 6! 9) = (9 6! + 5! 4!) 3! = ! = 8! = 3!! + ( ! 9) = ! + 5 3!!. 10
11 39600 = 3!! ( !) + 0 = 0 + (0! + 6 9) 3!! = (0! + 0! + 3!)! ( 6 + 9)!! = (9 6)!! + (3! + 0! + 0!)! = 3!! ( !) + 8 = 8 + (0! + 6 9) 3!! = 0! ( 3 + 6)!! + 8! + 9 = 9 + 8! (6 3)!! 0! = ( 3 + 9)! + (6 + 2)! + 4! = 4! + (2 + 6)! (9 3)! = (2 3 )! + 4! ( 6 + 9)!! = (9 6)!! + 4! + (3! + 2)! = 3!! + (9 6! + 3!) 6 = 6 (3! + 6! 9) + 3!! = 3!! + 3! (6 + 6! 9) = (9 6! + 6) 3! + 3!! = (3! + 9 7) 4! 4! = 4! 4! ( !) = ( 3!! + 4 4! + 7!) 9 = 9 (7! + 4 4! 3!!) = 3! + (9! 7!)/(5 + 4) = ((4 + 5)! 7!)/9 3! = 3! + ((4 + 5)! 7!)/9 = (9! 7!)/(5 + 4) 3! = ((3 + 6)! 7!)/9 + 3 = 3 + (9! 7!)/(6 + 3) = 3 + ((3 + 6)! 7!)/9 = (9! 7!)/(6 + 3) = ( 4! + (0! + (1 + 2)!)!) 8 = 8! + (2 10) 4! = ((0! + (1 + 2)!)! 4!) 8 = 8! + 4! (2 10) = (4 + 0!)! 1 + 9!/9 = 9!/9 1 (0! + 4)! = 0! (1 + 4)! + 9!/9 = 9!/9 (4 + 1)! 0! = 4 (0! + 22) + 8! = 8! (22 + 0!) 4 = (0! + 22) 4 + 8! = 8! 4 (22 + 0!) = ( 4 + 0!) ! = (8!/4! 2 0!) 4! = 02 4! 4! + 8! = 8! + 4! ( ) = ! 8 = 8! 7 ( ) = (02 + 4) 7 + 8! = 8 7! 42 0! = 4!/02 + 8! 4! = 4 + 8! 2 0!+4 = ( 0! 2 4) 4 + 8! = 8! = 4! (0! + 2)! + 8! 5 = 8! + 5 ( !) = (0! 2 4) 5 + 8! = 5 + 8! (2 + 0!)! 4! = 4! ! 7 = 7 + 8! 2 04! = 02 4! 7 + 8! = 8! 7 4! 2 0! = 4 (0 2 8) + 8! = 8! = ! = 8! = 4! ! 9 = 9 + 8! ! = 02 4! + 8! 9 = 9 + 8! 4! + 2 0! = 4! 0! + (2 (9 5))! = ((9 6) 2)! 0! 4! = 0! + (2 4)! ( 5 + 9)! = (9 5)! + (4 2)! 0! = 4! + 0! !/9 = 9!/ ! 4! = 0! + 2 4! + 9!/9 = 9!/9 4! ! = 4 (0 3) 0 + 8! = 8! (0 4) = ! = 8! 4 3 0! 0! = (4 0!)! + ( )! = (4 1)! + (3 + 0! + 4)! = 01 3! + (4 + 4)! = (4 + 4)! ( !)! = (40/(3! 1))! 5 = 5 + (13 0! 4)! = ( 0! )! 5 = 5 + (4 (3 1))! 0! = (1 + 7)! = (7 + 1)! = (0! !)! = 7 + (4!/3)! = ! = 8! = 0! ! = 8! = (40 32)! + 0 = 0 + ((2 + 30)/4)! = ((0! + (0 23)!) 4)! = (4!/3)! = (40 32)! + 1 = 1 + ((2 + 30)/4)! = ((0! + (0 23)!) 4)! = (4!/3)! = 0! + (12/3 + 4)! = ( )! = (40 32)! + 2 = 2 + ((2 + 30)/4)! = 02 + (2 3 )! + 4 = ( 4! + 32)! + 2 0! = (40 32)! + 3 = 3 + ((2 + 30)/4)! = 0! + ( )! + 4 = (4!/3)! = 4 (0 + 3) + (3! + 2)! = (2 3 )! + 3 (0 + 4) = (02 3 )! = (3! + 2 0!)! = 4! + ( 0! + 3 3)! 7 = 7 + (3 3 0!)! + 4! = ( 0! + 3 3)! + 4! 7 = 7 + 4! + (3 3 0!)!. 11
12 40338 = (4 + 0!) 3 + (3 + 8!) = 8! + 3! = ! + 8! = 8! = 4! + 0! 3 + (4 2)! = (2 4)! 3 + 0! + 4! = 0! (4 + 4)! = 4! + (4!/3)! 2 0! = 4! 0! + ((3! 4) 3 )! = ((3! 4) 3 )! 0! + 4! = 0! + (3! + 3! 4)! + 4! = 4! + (4!/3)! (3 0)! = 4! 0/3 + (4 + 4)! = (4 + 4)! + 3! (0 + 4) = (03 4 4)! + 4! = (4 + 4)! + 4 3! 0! = ! = 8! + (4 + 3) 04 = (03 + 4) 4 + 8! = 8! + 4 ( !) = ! = 8! ! + 4! = 0! ! = 8! ! = (( )! + 6) 8 = 8 (6 + ( )!) = 0! ! = 8! ! = 3! + 8! + 3 ( 0! + 4!) = (4! 0!) 3 + 8! 3! = 03 ( 3 + 4!) + 8! = 8! (3 0)! = 4! ! 6 = 6 + 8! ! = 03 4! 6 + 8! = 8! ! = 4 ( 0! + 4! 0!) + 8! = 8! + ( 0! + 4! 0!) 4 = (0! + 0!) ! = 8! + 44 (0! + 0!) = (4 (0! + (4! + 2))) + 8! = 8! + (2 + 4! + 0!) 4 = (0! !) 4 + 8! = 8! + 4 (4! !) = ( )! + (4 + 0!)! = 0 + (4 + 4)! + (0! + 4)! = ((0 0)! + 4)! + (4 + 4)! = (4 + 4)! + (4 0! + 0!)! = (4 + 0!)! + (4 + 4)! + 1 = 1 + (4 + 4)! + (0! + 4)! = 0! + (1 + 4)! + (4 + 4)! = (4 + 4)! + (4 + 1)! + 0! = (4 + 0!)! + (4 + 4)! + 4 = 4 + (4 + 4)! + (0! + 4)! = (0! + 4)! (4 + 4)! = 4 + (4 + 4)! + (4 + 0!)! = (4 + 0!)! + (4 + 4)! + 5 = 5 + (4 + 4)! + (0! + 4)! = 0! (4 + 4)! + 5! = 5 + (4 + 4)! + (4 + 0!)! = (4 + 0!)! + (4 + 4)! + 6 = 6 + (4 + 4)! + (0! + 4)! = (0! + 4)! + (4 + 4)! + 6 = 6 + (4 + 4)! + (4 + 0!)! = (4 + 0!)! + (4 + 4)! + 7 = 7 + (4 + 4)! + (0! + 4)! = (0! + 4)! + (4 + 4)! + 7 = 7 + (4 + 4)! + (4 + 0!)! = (4 + 0!)! + (4 + 4)! + 8 = 8 + (4 + 4)! + (0! + 4)! = (0! + 4)! ! = 8 + (4 + 4)! + (4 + 0!)! = (4 + 0!)! + (4 + 4)! + 9 = 9 + (4 + 4)! + (0! + 4)! = (0! + 4)! + (4 + 4)! + 9 = 9 + (4 + 4)! + (4 + 0!)! = (4 0!)! + 4! + 5! + 8! = 8! + 5! (4 0!)! + 4! = ( 0! + 4)! + 4! + 5! + 8! = 8! + 5! + 4! (4 0!)! = ( )! + 6 4! = 4! 6 + (4 + 04)! = (04 + 4)! + 4! 6 = 6 4! + ( )! = ! 6 + 8! = 8! + 6 4! + 04 = ! 6 + 8! = 8! + 6 4! + 4 0! = (4 + 0!)! + 5! + 8! + 4! = 4! + 8! + 5! + (0! + 4)! = (0! + 4)! + 4! + 5! + 8! = 8! + 5! + 4! + (4 + 0!)! = 4! + 0! + 5! + 8! + 5! = 5! + 8! + 5! + 0! + 4! = 0! + 4! + 5! + 5! + 8! = 8! + 5 (54 0!) = (4 0!)!! + 8! 4! 8 = 8 4! + 8! + ( 0! + 4)!! = ( 0! + 4)!! 4! 8 + 8! = 8! 8 4! + (4 0!)!! = 4 + ((1 + 0!) 3 )! + 6! = 6! + (3! + 0! + 1)! 4 = (0! !)! 4 + 6! = 6! 4 + (3! !)! = (4 1)!! + 0! 3 + 8! = 8! + 3!! + 0! = 0! !! 4 + 8! = 8! 4 + 3!! ! = (4 (1 + 0!))! + 6! + 4! = 4! + 6! + ((0! + 1) 4)! = ((0! + 1) 4)! + 4! + 6! = 6! + 4! + (4 (1 + 0!))! = 4! + 1 4! 4 /8 = 8! + 4! (4! 1 + 4!) = 1 4! + 4! 4 /8 = 8! + 4! (4! + 4! 1) = 4! + 1 4! 4 /8 = 8! + 4! (4! 1 + 4!) = 1 4! + 4! 4 /8 = 8! + 4! (4! + 4! 1). 12
13 41472 = 4! 1 4! 72 = 2 ( ) 4 = (1 + 2) 4! 4+7 = (7 4) 4! = (4 1)! 6 (1 + 6)! = (6 + 1)! ! = (11 4)! = 6 6 ( )! = (4 1)! ! = 7! ! = 1 + ( 1 + 4)! 6 7! = 7! ! = 4! 2 ( 0! + 4) + 8! = 8! + 4! (0! + 2) 4! = (0! + 2) 4! 4! + 8! = 8! + 4! 4! (2 + 0!) = (4 + 2) 3! 3! 6! = 63 (3!! 2 4!) = (2 3 )! + 3! 4 + 6! = !! + (3! + 2)! = (4 (3!! 1) + 8!) 8 = 8! 8 + ( 1 + 3!!) 4 = ( 1 + 3!!) ! = 8! (3!! 1) = 4 3!! 1 + 9!/9 = 9!/ !! 4 = 1 + 3!! 4 + 9!/9 = 9!/ !! = 4 (3!! + 2) 0 + 8! = 8! + ( !!) 4 = (02 + 3!!) 4 + 8! = 8! + 4 (3!! + 2 0!) = ((4! + 3) 5! + 8!) + 4! = 4! + 8! + 5! (3 + 4!) = 34 ( 4! + 5!) + 8! = 8! ! = 4 3! 6! + 8! 8 = ( )/3! 4 = 3!! ! = ( !)/3! = 4! 4 (5! 4) 4 = 4! 4 (5! 4) 4 = 4 4 4! ( 4 + 5!) = (5! 4) 4 4 4! = (4!/4) (6! 2) + 8! = 8! (2 6!) 4!/4 = ( 2 + (4!/4)!) 6 + 8! = 8! + 6 ((4!/4)! 2) = (4 + 4)! 6! 3 + 7! = 7! 3 6! + (4 + 4)! = 3 + (4 + 4)! 6! + 7! = 7! 6! + (4 + 4)! = (4 + 4)! + 6 (6! + 4) = (4 + 6!) 6 + (4 + 4)! = 4! + (4 + 4)! + 6 6! = 6 (6! + 4) + (4 + 4)! = 4! ! + 8! = 8! + 6 6! ! = 4!/4 (6! + 8) + 8! = 8! + (8 + 6!) 4!/ = 4! 4! + 7! + 8! 2 = 2 + 8! + 7! 4! 4! = 2! 4! 4! + 7! + 8! = 8! + 7! 4! 4! = ! = 8! = (2 + 4!) 4! 8 9 = 9 8 4! (4! + 2) = (4 + 5) ( !) = (7! 9 + 2) (5 + 4) = ( !) 9 = 9 (7! ) = (4 + 5) ((3 + 4)! 2) = ( 2 + (4 + 3)!) (5 + 4) = ( 2 + (3 + 4)!) (4 + 5) = (5 + 4) ((4 + 3)! 2) = ! 9 = 9 7! = 3! ! 9 = 9 (7! 5) = (4! 5 3)!/8 + 4! = 4! + 8! + (3! + 5 4)! = 3! 4 + (4 + 5)!/8 = (85 4!) (4! + 3!!) = (4 + 5) 3 + 8! + 7! = 7! + 8! + 3 (5 + 4) = 3 (4 + 5) + 7! + 8! = 8! + 7! + (5 + 4) = 4 + (5 + 6!) 7 9 = 9 7 (6! + 5) = 4 5! + 8! 3 + 7! = 7! 3 + 8! + 5! 4 = ! + 7! + 8! = 8! + 7! + 5! = 4! (5! (8 + 8) 8) = ( 8 + (8 + 8) 5!) 4! = (45 9) 3 6! = 6! + 3! (9 5)!/4 = (3!! 4!) (5! 6 9) = ( !) ( 4! + 3!!) = (4! + 6! + 2) 8 + 8! = 8! + 8 (2 + 6! + 4!) = (2 + 4! + 6! 8) 8 = 8! + 8 (6! + 4! + 2) = ! 6 = 6 3! = (3 4) 5! = 6 6 ( !)! = ( 4! + 6) = (6 4!) = 4! 5! = ! + 4! = 4! = ! = = = 4! = ! = 2 4! = ! 2. 13
14 46627 = 4! = ! = 2 4! = ! = 4! + (6 6) 3 1 = ! = 1 3 4! = ! = = (2 3) = (2 3) 4! = 6 6 4! (3 2) = = 3/ ! = ( 3 + 3)! 4! = 6 6 4! + ( 3 + 3)! = ! 4! = 4 + 3! ! = 3! = ! = ! 8 = 8 3! = 3! = ! = = = = = /4 8 = = = = = = 1 4 ( ) = = ! = 3! = 3 (4 5) = 6 6 (5 4) = = ! = (4 5) = = (( )/5) 6 = = ( 4 + 5) = 6 6 (6 5) = 4! = ! = ! = = 3! = ! = = = 4! = ! = 4! ! = 0! = 0! + 4! = ! + 0! = 4 + 6! + 7! 9 + 6! = 6! + 9 7! + 6! 4 = 4 + 6! + 6! + 7! 9 = 9 7! + 6! + 6! = 4 (8 + 9) 6! + 9 = 9 + 6! (9 8 4) = 4 6! (8 + 9) + 9 = 9 + (9 + 8) 6! = ((5 + 0!) 6 8!) 8 = 8 ( 8! + 6 0!+5 ) = ((0! + 5) 6 8!) 8 = 8 ( 8! + 6 (5+0!) ) = ( 5! + 0! + 7! + 6!) 9 = 9 (6! + 7! + 0! 5!) = (0! 5! + 6! + 7!) 9 = 9 (7! + 6! 5! + 0!) = (5! 3) ( 6 + 5!) + 8! = 8! + (5! 6) ( 3 + 5!) = ( 3 + 5!) (5! 6) + 8! = 8! (6 5!) (5! 3) = (5!/5) 2 96 = = = 96 (5!/5) = (5! + 6) 448 = 8! (4/4 + 6)/5 = ((4!/4) 5 6!) 8 = 8 (6 5 (4!/4)!) = (5! 6) (5! + 4) 4 = 4 (4 + 5!) ( 6 + 5!) = 4 (4 + 5!) (5! 6) = ( 6 + 5!) (5! + 4) = (5! 6) (5! + 4) 4 = 4 (4 + 5!) ( 6 + 5!) = 4 (4 + 5!) (5! 6) = ( 6 + 5!) (5! + 4) = 5 (7! + 6!) (0! + 0!) = (0! + 0!) (6! + 7!) 5 = (0! + 0!) 5 (6! + 7!) = (7! + 6!) 5 (0! + 0!) = ( !) 4! + 8! = 8! + 4! (6! + 7 5) = 4! ( 5 + 6! + 7) + 8! = 8! + (7 + 6! 5) 4! = ! + 5! 5! = 5! 5! + 7! 9 5 = 5! 5! 5 + 7! 9 = 9 7! 5 + 5! 5! = 6 0! + 4 7! 3 = 3 7! 4 0! 6 = ( 0! !) 7 = 7 (6! 4 3 0!). 14
15 60487 = (6 + 0!)! (4 + 8) + 7 = (0 + 6!) = (0! + 4 6!) 7 + 8! = 8! + 7 (6! 4 + 0!) = (6! 4 + 7! + 7!) 6 = (6! + 7! + 7! 4) 6 = 4! 6! + (6 + 7) 7! = 7! (7 + 6) 6! 4! = 6! (5! 5 4!) + 4! = 4! + ( 4! 5 + 5!) 6! = 4! (4! + 5 5!) 6! = 6! (5! 5 4!) + 4! = 6! 5! (6 + 6) 4 = ( 4! 6 + 6!) (5! 6) = ( 4! + 5!) ( !) = ( !) (5! 4!) = (7! 8) 4 = 4 ( 8 + 7!) = (3!! + 4) ! = 8! (4 + 3!!) = (6! + 9) ( !) = (5! 5 2 ) (9 + 6!) = ( !) (6! + 9) = (9 + 6!) (5! 5 2 ) = 6! + ( 9 + 7! + 7!) 7 = 7 (7! + 7! 9) 6! = 6! + 7 (7! + 7! 9) = ( 9 + 7! + 7!) 7 6! = 6! + 99 (6! 6) = (6! 6) 99 6! = 6! + ( 6 + 6!) 99 = 99 (6! 6) 6! = (7! 3 3!!) (8 + 9) = (9 + 8) (3!! + 3 7!) = ( 3!! 3 + 7!) (8 + 9) = (9 + 8) (7! 3 3!!) = ( 7! + 3!! 4!) 6 + 4! = 4! + 6 (4! 3!! 7!) = 3! (4 + 4! 6! 7!) = ( 7! + 6! 4! + 4) 3! = 7 4 (4! + 3! + 1) = (1 + 3!) 4 (4! + 7) = (7! 4! 6!) 8 + 8! = 8! 8 (6! + 4! 7!) = ( 4! 6! + 7!) 8 + 8! = 8! + 8 (7! 6! 4!) = (7! 5) (5 2) 5 = 5 ( 2 + 5) ( 5 + 7!) = ( 2 + 5) 5 ( 5 + 7!) = (7! 5) 5 (5 2) = (7! !) 3 = 3 ( 4! !) = ( 7 + 5! (5! + 6)) 5 = 5 ((6 + 5!) 5! 7) = 5 (5! 5! + 6! 7) = ( 7 + (6 + 5!) 5!) = 7! + (6! 3!) (5! 6) = ( 6 + 5!) ( 3! + 6!) 7! = ( 3! + 5!) ( 6 + 6!) 7! = 7! + ( 6 + 6!) (5! 3!) = ( 7 + 7! 6!) ( 3! + 4!) = (4! 3!) ( 6! 7 + 7!) = ( 3! + 4!) ( 6! 7 + 7!) = (7! 7 6!) (4! 3!) = (8! (0! + 4)!) (0! + 0!) = (0! + 0!)! ( (4 + 0!)! + 8!) = (0! + 0!)! ( (0! + 4)! + 8!) = (8! (4 + 0!)!) (0! + 0!) = 8! 0! 5! 1 + 8! = (8! 1 (5! + 0!)) + 8! = 0! 1 5! + 8! + 8! = 8! + 8! 5! 1 0! = (8! 0! 6 2) 2 = 2 ((2 + 6)! 0! 8) = (0! + 2)! + 2 ( 6 + 8!) = (8! 6) 2 (2 + 0!)! = (8! 0! 6) 2 3 = ( 6 0! + 8!) = 0! + 2 ( !) = (8! 6 3) 2 + 0! = (8! 0! 6) 2 3 = ( 6 0! + 8!) = 0! + 2 ( !) = (8! 6 3) 2 + 0! = 8! ! = 8! ! = ! + 8! = 8! + 8! 6 2 0! = (8! 0!) 6/3 4 = (4!/3)! ! = 03! + ( 4 + 6) 8! = 8! (6 4) 3! 0! = (5 + 3)! 6 + 0! + 8! = 8! (0 + 6/3) 5 = 0! + (3 + 5)! 6 + 8! = 8! 6 + (5 + 3)! + 0! = 8! (0 + 6/3) 5 = (5 + 3)! 6 + 0! + 8! = 0! + (3 + 5)! 6 + 8! = 8! 6 + (5 + 3)! + 0! = 8! + 0! ! = 8! ! + 8! = 0! ! + 8! = 8! + 8! ! = 8! + 0 6/3 + 8! = 8! ! + 8! = 0! ! + 8! = 8! + 8! 6/3 0! = 8! 0! + (6 + 3)!/9 = 9!/3 (6 0)! 8! = 0! !/9 = 9! 8/(6 3!) 0! = 8! ( ) + 7 = 7 + ( 4 + 6) (0 + 8!) = (0! + 4! 6!) 7 8! = 8! + 7 (6! 4! + 0!) = (8! + (0/6)! + 5) 2 = 2 (5 + (6 0)! + 8!) = ( 0! 2 + 5) (6 + 8!) = (8! + 6) ((5 2) 0!) = 8! ! = 8 (0 + 6) + (8! + 8!) = ! + 8! = 8! + 8! !. 15
16 81360 = 8! ( 1 + 3) + 6! + 0 = 0 + 6! + (3 1) 8! = (0! !)! + 6! + 8! = 8! + 6! + (3! !)! = 8! + (1 3)!! + 8! + 4! = 4! + 8! + 3!! + 1 8! = 1 3!! + 4! + 8! + 8! = 8! + 8! + 4! = ! 4! = 4! 4! = 2 4! 4! 8 9 = 9 8 4! = 8 + (3! + 1)! = 7! ! 8 = (1 + 3!)! = 8 + 7! + 5 3! = ( 4! + 3!!) 5! + 3! + 8 = 8 + 3! + 5! (3!! 4!) = 3! + 3!! ( 4 + 5!) + 8 = 8 + (5! 4) 3!! + 3! = 8 + 5! + 5! (6! 8) = ( 8 + 6!) 5! + 5! + 8 = 5! + 5! (6! 8) + 8 = 8 + ( 8 + 6!) 5! + 5! = 8 + 5! (6! 6) 4! = 4! + (6! 6) 5! + 8 = 4! + 5! ( 6 + 6!) + 8 = 8 + ( 6 + 6!) 5! 4! = 8 + 5! (6! 6) 6 = 6 + (6! 6) 5! 8 = 5! ( 6 + 6!) 6 8 = ( 6 + 6!) 5! = 8! ! 9 = 9 7! ! = ! (8 + 9) = (9 + 8) 7! = 8! + 5! + 7! 9 5 = 5! + 9 7! 5 + 8! = 5! 5 + 7! (8 + 9) = (9 + 8) 7! 5 + 5! = ( 8 + 6!) (1 + 5!) 1 = 1 + (5! + 1) (6! 8) = 1 + (1 + 5!) (6! 8) = ( 8 + 6!) (5! + 1) = ( 8 + 6!) ( 1 + 5! + 2) = (2 + 5! 1) (6! 8) = ( !) (6! 8) = ( 8 + 6!) (5! + 2 1) = 8! 6! + ( 2 + 5)! 6 = 6 5 2! 6! + 8! = ( 2 + 5)!! ! = 8! (5 2)!! = !! 5! 1 = 1 + 5! 3!! 6 8 = 1 + 3!! 5! 6 8 = ! 3!! = 8 6 3! + 5! 6! = 6! 5! 36 8 = 3!! 5! = ! 3!! = 8 6! (3! + 9) 5 = 5 + (9 + 3!) 6! 8 = 3!! 5! = ! 3!! = 8! + 6! ! = 7! 9 + ( 3 + 6!) + 8! = 3 + 6! + 7! (8 + 9) = (9 + 8) 7! + 6! = (8! + 6! 4) (0! + 0!) = (0! + 0!) (4 6! + 8!) = (8 + 6! 4!) (4 + 0!) = (0! + 4) (4! 6! + 8) = 8 + (6! 5 + 8) 4! = 4! ( !) 8 = 4! (5 6! + 8) 8 = 8 + (8 + 6! 5) 4! = 8 + 6!/6 (3!! + 2) = (2 + 3!!) 6!/ = ( !) (4 + 0!)! = (0! + 4)! (6! 6 + 8) = 8! + (6/(9 7))! 6 = 6 7 /(9 6)! + 8! = ( )! = ( )! = 9 ((0 + 7)! 0!) 2 = 2 ((0 + 7)! 0!) 9 = ( 0! 0! + 2 7!) 9 = 9 (7! 2 0! 0!) = 9 ( 0! + 7! (1 + 1)) = ((1 + 1) 7! 0!) 9 = ( 0! + (1 + 1) 7!) 9 = 9 (7! (1 + 1) 0!) = 9 (0 + 7)! = ! (0 + 9) = ( ) 7! 9 = 9!/(7 2 0! 0!) = 9 (0 + 7)! = ! (0 + 9) = ! 9 = 9!/(7 2 1) + 0! = 9 (0 + 7)! = ! (0 + 9) = ! 9 = 9 7! ! = 9 (0 + 7)! = ! (0 + 9) = 0! + 2 (7! + 8!) + 9 = 9 + (8! + 7!) 2 0! = ! (0 + 9) = 9 (0 + 7)! = 02 7! = ! 2 0! = (9 (0 + 7)! + 3!) 2 = 2 (3! + 7! ( 0 + 9)) = 02 (3! + 7! 9) = (9 7! + 3!) 2 0! = 9 (0! + 7!) ( 3! + 8) = (8 3!) (7! + 0!) 9 = ( 0! + 3) (7! + 8! + 9) = (9 + 8! + 7!) (3 0!). 16
17 90747 = 9 ( 0! + 7! !) = (7! ! 0!) 9 = ( 0! ! + 7!) 9 = 9 (7! + 7! + 4 0!) = 9 ( 0! + 7! !) = (7! ! 0!) 9 = ( 0! ! + 7!) 9 = 9 (7! + 7! + 4 0!) = 9 + 3! 3!+0! /3 = (3 0!) 3! 3! 9 = ( 0! + 3) 3! 3! 9 = 9 + 3! 3! (3 0!) = (9 3) 3! (1 2) = 2 1 3! 3+9 = ! 9 = (9 3) 3! = 9 + 3! 3! 2 0! = 0! + 2 3! 3! = 9 + 3! 3! 2 1 = 1 2 3! 3! = 9 + 3! 3! 2 1 = 1 2 3! 3! = (9 + 3! 3! ) (3 0!) = ( 0! + 3) (3! 3! + 9) = ( 9 + 3! 3! + 4!) 2 = 2 (4! + 3! 3! 9) = 2 (3! 3! + 4! 9) = ( 9 + 4! + 3! 3! ) = ( !) ( 0! + 3!!) = (3!! 0!) (5! ) = ( 0! + 3!!) (5! ) = ( !) (3!! 0!). 4 Sequential Representations of Selfie Numbers 4.1 Sequential Representations in Both Ways - In Order of Digits and Reverse 2160 = (2 + 1) 6! + 0 = 0 + 6! (1 + 2) = (2 + 1) 6! + 1 = 1 + 6! (1 + 2) = (2 + 1) 6! + 2 = 2 + 6! (1 + 2) = (2 + 1) 6! + 3 = 3 + 6! (1 + 2) = (2 + 1) 6! + 4 = 4 + 6! (1 + 2) = (2 + 1) 6! + 5 = 5 + 6! (1 + 2) = (2 + 1) 6! + 6 = 6 + 6! (1 + 2) = (2 + 1) 6! + 7 = 7 + 6! (1 + 2) = (2 + 1) 6! + 8 = 8 + 6! (1 + 2) = (2 + 1) 6! + 9 = 9 + 6! (1 + 2) = (2 + 5)!/2 + 0 = 0 + (2 + 5)!/ = (2 + 5)!/2 + 1 = 1 + (2 + 5)!/ = (2 + 5)!/2 + 2 = 2 + (2 + 5)!/ = (2 + 5)!/2 + 3 = 3 + (2 + 5)!/ = (2 + 5)!/2 + 4 = 4 + (2 + 5)!/ = (2 + 5)!/2 + 5 = 5 + (2 + 5)!/ = (2 + 5)!/2 + 6 = 6 + (2 + 5)!/ = (2 + 5)!/2 + 7 = 7 + (2 + 5)!/ = (2 + 5)!/2 + 8 = 8 + (2 + 5)!/ = (2 + 5)!/2 + 9 = 9 + (2 + 5)!/ = 3!! (6 0!) + 0 = 0 + ( 0! + 6) 3!! = 3!! (6 0!) + 1 = 1 + ( 0! + 6) 3!! = 3!! (6 0!) + 2 = 2 + ( 0! + 6) 3!! = 3!! (6 0!) + 3 = 3 + ( 0! + 6) 3!! = 3!! (6 0!) + 4 = 4 + ( 0! + 6) 3!! = 3!! (6 0!) + 5 = 5 + ( 0! + 6) 3!! = 3!! (6 0!) + 6 = 6 + ( 0! + 6) 3!! = 3!! (6 0!) + 7 = 7 + ( 0! + 6) 3!! = 3!! (6 0!) + 8 = 8 + ( 0! + 6) 3!! = 3!! (6 0!) + 9 = 9 + ( 0! + 6) 3!! = 5! + (1 + 6)! + 0 = 0 + (6 + 1)! + 5! = 5! + (1 + 6)! + 1 = 1 + (6 + 1)! + 5! = 5! + (1 + 6)! + 2 = 2 + (6 + 1)! + 5! = 5! + (1 + 6)! + 3 = 3 + (6 + 1)! + 5! = 5! + (1 + 6)! + 4 = 4 + (6 + 1)! + 5! = 5! + (1 + 6)! + 5 = 5 + (6 + 1)! + 5! = 5! + (1 + 6)! + 6 = 6 + (6 + 1)! + 5! = 5! + (1 + 6)! + 7 = 7 + (6 + 1)! + 5! = 5! + (1 + 6)! + 8 = 8 + (6 + 1)! + 5! = 5! + (1 + 6)! + 9 = 9 + (6 + 1)! + 5! = (1+0!) ( 0!+8)!+0 = 0+(8 0!)! (0!+1) = (1+0!) ( 0!+8)!+1 = 1+(8 0!)! (0!+1) = (1+0!) ( 0!+8)!+2 = 2+(8 0!)! (0!+1) = (1+0!) ( 0!+8)!+3 = 3+(8 0!)! (0!+1) = (1+0!) ( 0!+8)!+4 = 4+(8 0!)! (0!+1) = (1+0!) ( 0!+8)!+5 = 5+(8 0!)! (0!+1) = (1+0!) ( 0!+8)!+6 = 6+(8 0!)! (0!+1) = (1+0!) ( 0!+8)!+7 = 7+(8 0!)! (0!+1) = (1+0!) ( 0!+8)!+8 = 8+(8 0!)! (0!+1) = (1+0!) ( 0!+8)!+9 = 9+(8 0!)! (0!+1) = ! + 0 = 0 + 6! = ! 1 = 1 + 6! = ! 2 = 2 + 6! = ! + 3 = 3 + 6! = ! + 4 = 4 + 6! = ! + 5 = 5 + 6! = ! + 6 = 6 + 6! = ! + 7 = 7 + 6! = ! + 8 = 8 + 6! = ! + 9 = 9 + 6! = (1+4)! (4+0!)!+0 = 0+(0!+4)! (4+1)! = (1+4)! (4+0!)!+1 = 1+(0!+4)! (4+1)! = (1+4)! (4+0!)!+2 = 2+(0!+4)! (4+1)! = (1+4)! (4+0!)!+3 = 3+(0!+4)! (4+1)! = (1+4)! (4+0!)!+4 = 4+(0!+4)! (4+1)! = (1+4)! (4+0!)!+5 = 5+(0!+4)! (4+1)! = (1+4)! (4+0!)!+6 = 6+(0!+4)! (4+1)! = (1+4)! (4+0!)!+7 = 7+(0!+4)! (4+1)!. 17
18 14408 = (1+4)! (4+0!)!+8 = 8+(0!+4)! (4+1)! = (1+4)! (4+0!)!+9 = 9+(0!+4)! (4+1)! = 3! (0! )! + 0 = !! = 3! (0! )! + 1 = !! = 3! (0! )! + 2 = !! = 3! (0! )! + 3 = !! = 3! (0! )! + 4 = !! = 3! (0! )! + 5 = !! = 3! (0! )! + 6 = !! = 3! (0! )! + 7 = !! = 3! (0! )! + 8 = !! = 3! (0! )! + 9 = !! = (3 + 45) 6! + 0 = 0 + 6! (5 + 43) = (3 + 45) 6! + 1 = 1 + 6! (5 + 43) = (3 + 45) 6! + 2 = 2 + 6! (5 + 43) = (3 + 45) 6! + 3 = 3 + 6! (5 + 43) = (3 + 45) 6! + 4 = 4 + 6! (5 + 43) = (3 + 45) 6! + 5 = 5 + 6! (5 + 43) = (3 + 45) 6! + 6 = 6 + 6! (5 + 43) = (3 + 45) 6! + 7 = 7 + 6! (5 + 43) = (3 + 45) 6! + 8 = 8 + 6! (5 + 43) = (3 + 45) 6! + 9 = 9 + 6! (5 + 43) = 3!! (5 + 2) + 8! + 0 = 0 + 8! (2 5 3)! = 3!! (5 + 2) + 8! + 1 = 1 + 8! (2 5 3)! = 3!! (5 + 2) + 8! + 2 = 2 + 8! (2 5 3)! = 3!! (5 + 2) + 8! + 3 = 3 + 8! (2 5 3)! = 3!! (5 + 2) + 8! + 4 = 4 + 8! (2 5 3)! = 3!! (5 + 2) + 8! + 5 = 5 + 8! (2 5 3)! = 3!! (5 + 2) + 8! + 6 = 6 + 8! (2 5 3)! = 3!! (5 + 2) + 8! + 7 = 7 + 8! (2 5 3)! = 3!! (5 + 2) + 8! + 8 = 8 + 8! (2 5 3)! = 3!! (5 + 2) + 8! + 9 = 9 + 8! (2 5 3)! = 3!! ( !) + 0 = 0 + (4! + 4 7) 3!! = 3!! ( !) + 1 = 1 + (4! + 4 7) 3!! = 3!! ( !) + 2 = 2 + (4! + 4 7) 3!! = 3!! ( !) + 3 = 3 + (4! + 4 7) 3!! = 3!! ( !) + 4 = 4 + (4! + 4 7) 3!! = 3!! ( !) + 5 = 5 + (4! + 4 7) 3!! = 3!! ( !) + 6 = 6 + (4! + 4 7) 3!! = 3!! ( !) + 7 = 7 + (4! + 4 7) 3!! = 3!! ( !) + 8 = 8 + (4! + 4 7) 3!! = 3!! ( !) + 9 = 9 + (4! + 4 7) 3!! = 3!! (9 4)!+8!+0 = 0+8! ( 4+9)! 3!! = 3!! (9 4)!+8!+1 = 1+8! ( 4+9)! 3!! = 3!! (9 4)!+8!+2 = 2+8! ( 4+9)! 3!! = 3!! (9 4)!+8!+3 = 3+8! ( 4+9)! 3!! = 3!! (9 4)!+8!+4 = 4+8! ( 4+9)! 3!! = 3!! (9 4)!+8!+5 = 5+8! ( 4+9)! 3!! = 3!! (9 4)!+8!+6 = 6+8! ( 4+9)! 3!! = 3!! (9 4)!+8!+7 = 7+8! ( 4+9)! 3!! = 3!! (9 4)!+8!+8 = 8+8! ( 4+9)! 3!! = 3!! (9 4)!+8!+9 = 9+8! ( 4+9)! 3!! = 3!! ( !) + 0 = 0 + (0! + 6 9) 3!! = 3!! ( !) + 1 = 1 + (0! + 6 9) 3!! = 3!! ( !) + 2 = 2 + (0! + 6 9) 3!! = 3!! ( !) + 3 = 3 + (0! + 6 9) 3!! = 3!! ( !) + 4 = 4 + (0! + 6 9) 3!! = 3!! ( !) + 5 = 5 + (0! + 6 9) 3!! = 3!! ( !) + 6 = 6 + (0! + 6 9) 3!! = 3!! ( !) + 7 = 7 + (0! + 6 9) 3!! = 3!! ( !) + 8 = 8 + (0! + 6 9) 3!! = 3!! ( !) + 9 = 9 + (0! + 6 9) 3!! = 3!!/9 6! + 8! + 0 = 0 + 8! + 6!/9 3!! = 3!!/9 6! + 8! + 1 = 1 + 8! + 6!/9 3!! = 3!!/9 6! + 8! + 2 = 2 + 8! + 6!/9 3!! = 3!!/9 6! + 8! + 3 = 3 + 8! + 6!/9 3!! = 3!!/9 6! + 8! + 4 = 4 + 8! + 6!/9 3!! = 3!!/9 6! + 8! + 5 = 5 + 8! + 6!/9 3!! = 3!!/9 6! + 8! + 6 = 6 + 8! + 6!/9 3!! = 3!!/9 6! + 8! + 7 = 7 + 8! + 6!/9 3!! = 3!!/9 6! + 8! + 8 = 8 + 8! + 6!/9 3!! = 3!!/9 6! + 8! + 9 = 9 + 8! + 6!/9 3!! = (40 32)! + 0 = 0 + ((2 + 30)/4)! = (40 32)! + 1 = 1 + ((2 + 30)/4)! = (40 32)! + 2 = 2 + ((2 + 30)/4)! = (40 32)! + 3 = 3 + ((2 + 30)/4)! = (40 32)! + 4 = 4 + ((2 + 30)/4)! = (40 32)! + 5 = 5 + ((2 + 30)/4)! = (40 32)! + 6 = 6 + ((2 + 30)/4)! = (40 32)! + 7 = 7 + ((2 + 30)/4)! = (40 32)! + 8 = 8 + ((2 + 30)/4)! = (40 32)! + 9 = 9 + ((2 + 30)/4)! = (4+0!)!+(4+4)!+0 = 0+(4+4)!+(0!+4)! = (4+0!)!+(4+4)!+1 = 1+(4+4)!+(0!+4)! = (4+0!)!+(4+4)!+2 = 2+(4+4)!+(0!+4)! = (4+0!)!+(4+4)!+3 = 3+(4+4)!+(0!+4)! = (4+0!)!+(4+4)!+4 = 4+(4+4)!+(0!+4)! = (4+0!)!+(4+4)!+5 = 5+(4+4)!+(0!+4)! = (4+0!)!+(4+4)!+6 = 6+(4+4)!+(0!+4)! = (4+0!)!+(4+4)!+7 = 7+(4+4)!+(0!+4)! = (4+0!)!+(4+4)!+8 = 8+(4+4)!+(0!+4)! = (4+0!)!+(4+4)!+9 = 9+(4+4)!+(0!+4)! = ! + 0 = 0 + 8! = ! + 1 = 1 + 8! = ! + 2 = 2 + 8! = ! + 3 = 3 + 8! = ! + 4 = 4 + 8! = ! + 5 = 5 + 8! = ! + 6 = 6 + 8! = ! + 7 = 7 + 8! = ! + 8 = 8 + 8! = ! + 9 = 9 + 8! = 4 (8 + 9) 6! + 0 = 0 + 6! (9 8 4) = 4 (8 + 9) 6! + 1 = 1 + 6! (9 8 4) = 4 (8 + 9) 6! + 2 = 2 + 6! (9 8 4) = 4 (8 + 9) 6! + 3 = 3 + 6! (9 8 4) = 4 6! (9 + 8) + 4 = 4 (8 + 9) 6!
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