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List the expressionist numbers (Posted on 2016-05-30) Difficulty: 4 of 5
Consider numbers like 343 or 2592.
343=(3+4)^3
2592=2^5*9^2.

We have just shown that some numbers stay unchanged when a number of mathematical operators are added /inserted, without changing the order of the digits.
Within our puzzle lets call those numbers expressionist numbers.
Let us limit the set of acceptable mathematical symbols to the following operators: +, -, *, /, ^, sqrt, !. and any amount of brackets.

My questions:
In the period between 10 A.D. & 2016 A.D. what years were labeled by expressionist numbers?
How many such years will there be between 2017 A.D. & 9999 A.D.?

Bonus: How about one, two (or more) 5-digit examples?

No Solution Yet Submitted by Ady TZIDON    
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Solution Part 1 redone allowing unary negation | Comment 9 of 10 |
I realized that allowing unary negation only at the beginning, such as -2+9 for 7, was insufficient. Such situations can be needed in the middle of a formula.

However the program I used contains a bug causing some spurious results, such as for 1775 below.  I will rerun the corrected program over tonight.

Allowing for this really slows down the processing, so I'd hate to do it for numbers higher than 2016. But for part 1, the list has expanded from 55 to 82, but the formulas are more complex, sometimes unnecessarily so due to the algorithm used:

24   2n4r^!                     ((-(2))^sqrt(4))!
36   3!6*                       (3)!*6
71   7!1+r                      sqrt(((7)!+1))
115  1n1n5!+n*                  (-(1))*(-((-(1)+(5)!)))
119  1n1n9r!+!+                 -(1)+((-(1)+(sqrt(9))!))!
120  1n2n0!-n!+!                ((-(1)+((-((-(2)-(0)!))))!))!
125  1n2rrn5!-*                 (-(1))*(-(sqrt(sqrt(2)))-(5)!)
127  1n2n7^-                    -(1)-(-(2))^7
143  1n4!n3!*-                  -(1)-(-((4)!))*(3)!
144  1n4+!4!*                   ((-(1)+4))!*(4)!
145  14!+5!+                    1+(4)!+(5)!
216  21+!r6^                    sqrt(((2+1))!)^6
240  2n4!n0!-nr!n*              (-(2))*(-((sqrt((-((-((4)!)-(0)!)))))!))
254  2n5rrn4^+                  -(2)+(-(sqrt(sqrt(5))))^4
324  3!2r/4^                    ((3)!/sqrt(2))^4
343  3n4n+3!^r                  sqrt((-(3)+-(4))^(3)!)
345  3n4rrn5!+n*                (-(3))*(-((-(sqrt(sqrt(4)))+(5)!)))
354  3n5!n4r+*                  (-(3))*(-((5)!)+sqrt(4))
355  3n5!n*5n+                  (-(3))*(-((5)!))+-(5)
360  3n6n0!+n!n*                (-(3))*(-(((-((-(6)+(0)!))))!))
375  3n7rrn5!-*                 (-(3))*(-(sqrt(sqrt(7)))-(5)!)
384  3!rr8r*4^                  (sqrt(sqrt((3)!))*sqrt(8))^4
456  4n5!n6+*                   (-(4))*(-((5)!)+6)
575  5n7rrn5!+n*                (-(5))*(-((-(sqrt(sqrt(7)))+(5)!)))
595  5n9rrn5!*-                 -(5)-(-(sqrt(sqrt(9))))*(5)!
605  6!0!n5!+-                  (6)!-(-((0)!)+(5)!)
624  6!2rrn4!*+                 (6)!+(-(sqrt(sqrt(2))))*(4)!
639  6!3rrn9n*-                 (6)!-(-(sqrt(sqrt(3))))*(-(9))
693  6!9rrn3!^-                 (6)!-(-(sqrt(sqrt(9))))^(3)!
695  6!9rrn5n*-                 (6)!-(-(sqrt(sqrt(9))))*(-(5))
713  7n1n3!!*-                  -(7)-(-(1))*((3)!)!
715  71n+!5n+                   ((7+-(1)))!+-(5)
720  7n2n+0!n*r!!               ((sqrt(((-(7)+-(2))*(-((0)!)))))!)!
721  72+r!!1+                   ((sqrt((7+2)))!)!+1
722  72+r!!2+                   ((sqrt((7+2)))!)!+2
723  72+r!!3+                   ((sqrt((7+2)))!)!+3
724  72+r!!4+                   ((sqrt((7+2)))!)!+4
725  72+r!!5+                   ((sqrt((7+2)))!)!+5
726  72+r!!6+                   ((sqrt((7+2)))!)!+6
727  72+r!!7+                   ((sqrt((7+2)))!)!+7
728  72+r!!8+                   ((sqrt((7+2)))!)!+8
729  7n2n+9r!^r                 sqrt((-(7)+-(2))^(sqrt(9))!)
733  73!!+3!+                   7+((3)!)!+(3)!
736  73n6^+                     7+(-(3))^6
744  7n4!-4!n*                  (-(7)-(4)!)*(-((4)!))
936  9r!r3!^6!+                 sqrt((sqrt(9))!)^(3)!+(6)!
1024 1n0!-2rrn4!+^r             sqrt((-(1)-(0)!)^(-(sqrt(sqrt(2)))+(4)!))
1285 1n2n8^-5n*                 (-(1)-(-(2))^8)*(-(5))
1288 12n-!r8^8n+                sqrt(((1-(-(2))))!)^8+-(8)
1294 1n2n9r!n4^+n*              (-(1))*(-((-(2)+(-((sqrt(9))!))^4)))
1296 1n2rrn9rrn6n^r/*           (-(1))*(-(sqrt(sqrt(2))))/sqrt((-(sqrt(sqrt(9))))^(-(6)))
1298 1n2n*9r!rn8^+              (-(1))*(-(2))+(-(sqrt((sqrt(9))!)))^8
1392 13n-!9r!!-2n*              (((1-(-(3))))!-((sqrt(9))!)!)*(-(2))
1432 1n4n*3!!-2n*               ((-(1))*(-(4))-((3)!)!)*(-(2))
1433 1n4rn3rrn3!!+*-            -(1)-(-(sqrt(4)))*(-(sqrt(sqrt(3)))+((3)!)!)
1434 1n4rrn3!!+-4rn*            (-(1)-(-(sqrt(sqrt(4)))+((3)!)!))*(-(sqrt(4)))
1435 1n4r*3!!n*5n+              (-(1))*sqrt(4)*(-(((3)!)!))+-(5)
1436 1n4n3!!+n*6!+              (-(1))*(-((-(4)+((3)!)!)))+(6)!
1439 1n4rrn3n-n!+9r!!+          -(1)+((-((-(sqrt(sqrt(4)))-(-(3))))))!+((sqrt(9))!)!
1440 1n4r*4rn0!-n!!n*           (-(1))*sqrt(4)*(-((((-((-(sqrt(4))-(0)!))))!)!))
1441 1n4n-!!4r*1n-              (((-(1)-(-(4))))!)!*sqrt(4)-(-(1))
1442 1n4rrn4n+n!-2n*            (-(1)-((-((-(sqrt(sqrt(4)))+-(4)))))!)*(-(2)) 
1443 1n4rrn4rn3!!*+*            (-(1))*(-(sqrt(sqrt(4)))+(-(sqrt(4)))*((3)!)!) 
1444 1n4n-!!4r+4r*              ((((-(1)-(-(4))))!)!+sqrt(4))*sqrt(4) 
1445 1n4n-!!4r*5n-              (((-(1)-(-(4))))!)!*sqrt(4)-(-(5)) 
1446 1n4rrn4rn6!*+*             (-(1))*(-(sqrt(sqrt(4)))+(-(sqrt(4)))*(6)!) 
1447 1n4n-!!4r*7n-              (((-(1)-(-(4))))!)!*sqrt(4)-(-(7)) 
1448 1n4n-!!4r*8n-              (((-(1)-(-(4))))!)!*sqrt(4)-(-(8)) 
1449 1n4rrn4rn9r!!*+*           (-(1))*(-(sqrt(sqrt(4)))+(-(sqrt(4)))*((sqrt(9))!)!) 
1459 14rrn5^nrn9r!*-            1-(-(sqrt((-((-(sqrt(sqrt(4))))^5)))))*(sqrt(9))! 
1463 1n4!+6!+3!!+               -(1)+(4)!+(6)!+((3)!)! 
1464 1n4r*6!n*4!+               (-(1))*sqrt(4)*(-((6)!))+(4)! 
1573 1n5!-7n3!-*                (-(1)-(5)!)*(-(7)-(3)!) 
1673 1n6n+7!n3n/+               -(1)+-(6)+(-((7)!))/(-(3)) 
1679 1n6!n7n*n9r/-              -(1)-(-(((-((6)!))*(-(7)))))/sqrt(9) 
1680 16n-!8n0!-nr/              ((1-(-(6))))!/sqrt((-((-(8)-(0)!)))) 
1684 1n6rrn8!n4!/+*             (-(1))*(-(sqrt(sqrt(6)))+(-((8)!))/(4)!) 
1704 1n7!-0!n*r4!*              sqrt(((-(1)-(7)!)*(-((0)!))))*(4)! 
1764 1n7r*6r*4^                 ((-(1))*sqrt(7)*sqrt(6))^4 
1775 17!+r7rrn5n**              sqrt((1+(7)!))*(-(sqrt(sqrt(7))))*(-(5)) 
1915 19r!+!1n5^+                ((1+(sqrt(9))!))!+(-(1))^5 
1944 1n9r!rr*4!^4!/             ((-(1))*sqrt(sqrt((sqrt(9))!)))^(4)!/(4)! 

Edited on June 1, 2016, 8:29 pm
  Posted by Charlie on 2016-06-01 08:04:18

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