perplexus.info :: Numbers : List the expressionist numbers

List the expressionist numbers (Posted on 2016-05-30)

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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Part 2 and Bonus

| Comment 4 of 10 |

(In reply to Computer "solution" part 1 -- no guarantee of completeness by Charlie)

I count 188 cases below for part 2:

2048  2r0!4!+^8r/   sqrt(2)^(0!+4!)/sqrt(8)
2160  21+6!*        (2+1)*6!                   should have +0   (0+ in RPN)
2161  21+6!*1+      (2+1)*6!+1 
2162  21+6!*2+      (2+1)*6!+2
2163  21+6!*3+      (2+1)*6!+3
2164  21+6!*4+      (2+1)*6!+4
2165  21+6!*5+      (2+1)*6!+5
2166  21+6!*6+      (2+1)*6!+6
2167  21+6!*7+      (2+1)*6!+7
2168  21+6!*8+      (2+1)*6!+8
2169  21+6!+9r*     (2+1+6!)*sqrt(9)
2187  21+r8^rr7^    sqrt(sqrt(sqrt((2+1))^8))^7
2304  2r30!+!r*4^   (sqrt(2)*sqrt((3+0!)!))^4
2520  25+!2/        (2+5)!/2                     should have +0   (0+ in RPN)
2521  25+!2/1+      (2+5)!/2+1
2522  25+!2/2+      (2+5)!/2+2
2523  25+!2/3+      (2+5)!/2+3
2524  25+!2/4+      (2+5)!/2+4
2525  25+!2/5+      (2+5)!/2+5
2526  25+!2/6+      (2+5)!/2+6
2527  25+!2/7+      (2+5)!/2+7
2528  25+!2/8+      (2+5)!/2+8
2529  25+!2/9+      (2+5)!/2+9
2544  25+!4r/4!+    (2+5)!/sqrt(4)+4!
2592  2r5^9*2^      (sqrt(2)^5*9)^2
2737  2r7r*3!^7-    (sqrt(2)*sqrt(7))^3!-7
2742  2r7r*r4!^r2-  sqrt(sqrt((sqrt(2)*sqrt(7)))^4!)-2
2744  2r7rr4r^*r4!^rsqrt(sqrt((sqrt(2)*sqrt(sqrt(7))^sqrt(4)))^4!)
2746  27r4rr*6^+    2+(sqrt(7)*sqrt(sqrt(4)))^6
2864  2r8r*6!4-*    sqrt(2)*sqrt(8)*(6!-4)
2880  2r8r*80!+r!!* sqrt(2)*sqrt(8)*sqrt((8+0!))!!
2896  289r!!+6!+*   2*(8+sqrt(9)!!+6!)
2904  29r+!0!+4!*   ((2+sqrt(9))!+0!)*4!
2954  29r5!+4!*+    2+(sqrt(9)+5!)*4!
3125  3!1+2-5^      (3!+1-2)^5
3249  3!!2+4r/9*    ((3!!+2)/sqrt(4))*9
3444  3!4!r4^4r-*   3!*(sqrt(4!)^4-sqrt(4))
3448  3!rr4!r*4^8-  (sqrt(sqrt(3!))*sqrt(4!))^4-8
3453  3!!4!*5/3-    3!!*4!/5-3
3454  3!!4!*5/4r-   3!!*4!/5-sqrt(4)
3455  3!!4!-5-5*    (3!!-4!-5)*5
3456  3!!r4!*5/6!r* (sqrt(3!!)*4!/5)*sqrt(6!)
3459  3!!4!*5/9r+   3!!*4!/5+sqrt(9)
3462  3!4!6r*2^+    3!+(4!*sqrt(6))^2
3495  3!!4!-9r+5*   (3!!-4!+sqrt(9))*5
3564  3!!5*6r4^-    3!!*5-sqrt(6)^4
3579  3!!5*79r*-    3!!*5-7*sqrt(9)
3584  3!!5*8+4!-    3!!*5+8-4!
3585  3!!5+8-5*     (3!!+5-8)*5
3586  3!!5*8-6-     3!!*5-8-6
3589  3!!5*8-9r-    3!!*5-8-sqrt(9)
3590  3!!5*9-0!-    3!!*5-9-0!
3591  3!!5*9-       3!!*5-9
3592  3!!5*9r!-2-   3!!*5-sqrt(9)!-2
3594  3!!5*9r!rr4^- 3!!*5-sqrt(sqrt(sqrt(9)!))^4
3594  3!!5*9r!-     3!!*5-sqrt(9)!
3595  3!!r5*9r!!r*5-sqrt(3!!)*5*sqrt(sqrt(9)!!)-5
3597  3!!5*9r-      3!!*5-sqrt(9)
3598  3!!5*9r!+8-   3!!*5+sqrt(9)!-8
3599  3!!5*9rr9rr-!-3!!*5-(sqrt(sqrt(9))-sqrt(sqrt(9)))!
3600  3!!60!-*      3!!*(6-0!)
3601  3!!60!-*1+    3!!*(6-0!)+1
3602  3!!60!-*2+    3!!*(6-0!)+2
3603  3!!60!-*3+    3!!*(6-0!)+3
3604  3!!6!0!+4*+   3!!+(6!+0!)*4
3605  3!!r6!r*0!+5* (sqrt(3!!)*sqrt(6!)+0!)*5
3606  3!!60!-*6+    3!!*(6-0!)+6
3607  3!!60!-*7+    3!!*(6-0!)+7
3608  3!!60!-*8+    3!!*(6-0!)+8
3609  3!!60!-*9+    3!!*(6-0!)+9
3615  36!+1*5*      (3+6!)*1*5
3625  36!+2+5*      (3+6!+2)*5
3630  3!!6+3!0!-*   (3!!+6)*(3!-0!)
3636  3!!6+3!*6!-   (3!!+6)*3!-6!
3645  3r6r^4!r^5*   sqrt(3)^sqrt(6)^sqrt(4!)*5
3654  3!!6+5*4!+    (3!!+6)*5+4!
3655  3!!6+5+5*     (3!!+6+5)*5
3685  36^8+5*       (3^6+8)*5
3755  3!!7+5*5!+    (3!!+7)*5+5!
3780  3!7!*8/       3!*7!/8                      should have +0   (0+ in RPN)
3781  3!7!*8/1+     3!*7!/8+1
3782  3!7!*8/2+     3!*7!/8+2
3783  3!7!*8/3+     3!*7!/8+3
3784  3!7!*8/4+     3!*7!/8+4
3785  3!7!*8/5+     3!*7!/8+5
3786  3!7!*8/6+     3!*7!/8+6
3787  3!7!*8/7+     3!*7!/8+7
3788  3!7!*8/8+     3!*7!/8+8
3789  3!7!*8/9+     3!*7!/8+9
  

At this point the maximum depth was changed to 16 to speed the process, which was slowing down considerably. 
The previous solutions did not need more than that anyway.
The conversions below were made with the RPN to Algebraic conversion program.

3972  397*2^+                 3+(9*7)^2
3996  3!!9r!9*-6*             (3!!-sqrt(9)!*9)*6
4032  4!0!3!+r*2^             (4!*sqrt((0!+3!)))^2
4088  40!8+r!^8-              4^sqrt((0!+8))!-8
4093  4rr0!9r+!^3-            sqrt(sqrt(4))^(0!+sqrt(9))!-3
4094  4rr0!9r+!^4r-           sqrt(sqrt(4))^(0!+sqrt(9))!-sqrt(4)
4096  4rr0!9r+!^              sqrt(sqrt(4))^(0!+sqrt(9))!
4098  4r0!9-8^r+              sqrt(4)+sqrt((0!-9)^8)
4099  4rr0!9r+!^9r+           sqrt(sqrt(4))^(0!+sqrt(9))!+sqrt(9)
4314  4!3!!1-*4/              4!*(3!!-1)/4
4316  4r3!!1-6*+              sqrt(4)+(3!!-1)*6
4319  43+!1-9r!!-             (4+3)!-1-sqrt(9)!!
4320  4r3!!*20!+*             sqrt(4)*3!!*(2+0!)
4324  43!!24+*+               4+3!!*(2+4)
4330  43!3!!0!+*+             4+3!*(3!!+0!)
4331  4r3!!+3!*1-             (sqrt(4)+3!!)*3!-1
4332  4r3!!+r3!r*2^           (sqrt((sqrt(4)+3!!))*sqrt(3!))^2
4334  4r3!!+3!*4r+            (sqrt(4)+3!!)*3!+sqrt(4)
4337  4!3!!3!*+7-             4!+3!!*3!-7
4344  4!3!!*4/4!+             4!*3!!/4+4!
4346  4r3!!4+6*+              sqrt(4)+(3!!+4)*6
4368  4r3*6!8+*               sqrt(4)*3*(6!+8)
4372  4r37^*2-                sqrt(4)*3^7-2
4374  4rr3r7^*r4^             sqrt((sqrt(sqrt(4))*sqrt(3)^7))^4
4464  4r4+6!4!+*              (sqrt(4)+4)*(6!+4!)
4466  4r4!6!+6*+              sqrt(4)+(4!+6!)*6
4480  4r4*!80!+/              (sqrt(4)*4)!/(8+0!)
4608  4!r6^0!8+r/             sqrt(4!)^6/sqrt((0!+8))
4816  481+r!^6!+              4^sqrt((8+1))!+6!
4896  4!rr8^9r!!6*+           sqrt(sqrt(4!))^8+sqrt(9)!!*6
4913  4r9*1-r3!^              sqrt((sqrt(4)*9-1))^3!
4944  49r+!4!4*-              (4+sqrt(9))!-4!*4
4995  49r+!95*-               (4+sqrt(9))!-9*5
5035  5rr0*!3!+!5-            ((sqrt(sqrt(5))*0)!+3!)!-5
5037  5rr0*3-7!+              sqrt(sqrt(5))*0-3+7!
5039  50!3!+!+9r!-            5+(0!+3!)!-sqrt(9)!
5040  5rr0*!4r+!0!+!          (((sqrt(sqrt(5))*0)!+sqrt(4))!+0!)!
5041  5!0!+r4-!1+             (sqrt((5!+0!))-4)!+1
5042  5!0!+r4-!2+             (sqrt((5!+0!))-4)!+2
5043  5!0!+r4-!3+             (sqrt((5!+0!))-4)!+3
5044  5!0!+r4-!4+             (sqrt((5!+0!))-4)!+4
5045  5!0!+r4-!5+             (sqrt((5!+0!))-4)!+5
5046  5!0!+r4-!6+             (sqrt((5!+0!))-4)!+6
5047  5!0!+r4-!7+             (sqrt((5!+0!))-4)!+7
5048  5!0!+r4-!8+             (sqrt((5!+0!))-4)!+8
5049  5!0!+r4-!9+             (sqrt((5!+0!))-4)!+9
5064  5rr0*!6+!4!+            ((sqrt(sqrt(5))*0)!+6)!+4!
5160  5!1*60!+!+              5!*1+(6+0!)!
5161  5!1+61+!+               5!+1+(6+1)!
5162  5!16+!+2+               5!+(1+6)!+2
5163  5!16+!+3+               5!+(1+6)!+3
5164  5!16+!+4+               5!+(1+6)!+4
5165  5!16+!+5+               5!+(1+6)!+5
5166  5!16+!+6+               5!+(1+6)!+6
5167  5!1+6+7!+               5!+1+6+7!
5168  5!16+!+8+               5!+(1+6)!+8
5169  5!16+!+9+               5!+(1+6)!+9
5184  51+r8^4*                sqrt((5+1))^8*4
5275  5!2*7!+5-               5!*2+7!-5
5280  5!2*80!-!+              5!*2+(8-0!)!
5397  5!3*9r-7!+              5!*3-sqrt(9)+7!
5765  57!+6!+                 5+7!+6!
5864  5!86!4r-*+              5!+8*(6!-sqrt(4))
5875  5!8!7/+5-               5!+8!/7-5
5880  5!8!80!-/+              5!+8!/(8-0!)
6399  6!r3!!r*9-9*            (sqrt(6!)*sqrt(3!!)-9)*9
6455  64^5-5*                 (6^4-5)*5
6475  6!4r*7!+5-              6!*sqrt(4)+7!-5
6476  6!4-7!+6!+              6!-4+7!+6!
6480  6rr4^!80!+*             sqrt(sqrt(6))^4!*(8+0!)
6494  6!4r+9*4-               (6!+sqrt(4))*9-4
6495  64^9r+5*                (6^4+sqrt(9))*5
6498  6!4r+rr9rr*r8^          sqrt((sqrt(sqrt((6!+sqrt(4))))*sqrt(sqrt(9))))^8
6696  6r6^96!*+               sqrt(6)^6+9*6!
6768  67!6/+8*                (6+7!/6)*8
6835  6!8!+3!/5-              (6!+8!)/3!-5
6839  6!8!+3!-9r!/            (6!+8!-3!)/sqrt(9)!
6840  6!8!+4r0!+!/            (6!+8!)/(sqrt(4)+0!)!
6859  68+5+r9r!^              sqrt((6+8+5))^sqrt(9)!
6864  6!8!+6/4!+              (6!+8!)/6+4!
7056  70!-5^6!-               (7-0!)^5-6!
7199  7!1-9r!!9r*+            7!-1+sqrt(9)!!*sqrt(9)
7235  723!!*+5*               (7+2*3!!)*5
7344  7!3r4*4^+               7!+(sqrt(3)*4)^4
7595  75!9*5+*                7*(5!*9+5)
7927  792+!7!/+               7+(9+2)!/7!
7944  7!9r!!4*+4!+            7!+sqrt(9)!!*4+4!
8057  8!0!*5/7-               8!*0!/5-7
8064  8!0!6rr^4!+r/           8!/sqrt((0!^sqrt(sqrt(6))+4!))
8065  8!0!-6+5/               (8!-0!+6)/5
8192  8r1*9^2r/               (sqrt(8)*1)^9/sqrt(2)
8397  8!3/9r-7!-              8!/3-sqrt(9)-7!
8405  8!4!/0!+5*              (8!/4!+0!)*5
8644  86!4!*+4r/              (8+6!*4!)/sqrt(4)
8648  86!48+*+                8+6!*(4+8)
8974  8!9/7+4r*               (8!/9+7)*sqrt(4)
9360  9r!!3!*60!+!+           sqrt(9)!!*3!+(6+0!)!
9595  9r!!5!*9/5-             sqrt(9)!!*5!/9-5
9599  9r!!5!*9-9/             (sqrt(9)!!*5!-9)/9
Bonus:
10074  10!+0!7!+4-*            (1+0!)*(0!+7!-4)
10075  10!+0!*7!*5-            (1+0!)*0!*7!-5
10076  10!+0!7!+*6-            (1+0!)*(0!+7!)-6
10079  10!+0!7!+*9r-           (1+0!)*(0!+7!)-sqrt(9)
10080  10!+0!*80!-!*           (1+0!)*0!*(8-0!)!
10081  10!0!+81-!*+            1+(0!+0!)*(8-1)!
10082  10!0!8+r!+!+2*          (1+(0!+(sqrt(0!+8))!)!)*2
10368  10!+r3!^rr6*r8^         (sqrt((sqrt(sqrt(sqrt(1+0!))^(3!)))*6))^8

RPN2ALGB.BAS:

DECLARE SUB checkParen (s$, ops$)

OPEN "rpn.TXT" FOR INPUT AS #1

OPEN "algbrc.TXT" FOR OUTPUT AS #2

DIM stack$(25)

LINE INPUT #1, l$

stackLevel = 0

bld$ = ""

FOR i = 1 TO LEN(l$)

c$ = MID$(l$, i, 1)

ix = INSTR("0123456789", c$)

IF ix THEN

stackLevel = stackLevel + 1

stack$(stackLevel) = c$

ELSE

IF stkRaise THEN

stackLevel = stackLevel + 1

stack$(stackLevel) = bld$

stkRaise = 0

bld$ = ""

END IF

SELECT CASE c$

CASE "+"

stackLevel = stackLevel - 1

stack$(stackLevel) = stack$(stackLevel) + "+" + stack$(stackLevel + 1)

CASE "-"

h2$ = stack$(stackLevel)

checkParen h2$, "+-"

stackLevel = stackLevel - 1

h1$ = stack$(stackLevel)

stack$(stackLevel) = h1$ + "-" + h2$

CASE "*"

h2$ = stack$(stackLevel)

checkParen h2$, "+-"

stackLevel = stackLevel - 1

h1$ = stack$(stackLevel)

checkParen h1$, "+-/"

stack$(stackLevel) = h1$ + "*" + h2$

CASE "/"

h2$ = stack$(stackLevel)

checkParen h2$, "+-/*"

stackLevel = stackLevel - 1

h1$ = stack$(stackLevel)

checkParen h1$, "+-/"

stack$(stackLevel) = h1$ + "/" + h2$

CASE "^"

h2$ = stack$(stackLevel)

checkParen h2$, "+-*/"

stackLevel = stackLevel - 1

h1$ = stack$(stackLevel)

checkParen h1$, "+-*/"

stack$(stackLevel) = h1$ + "^" + h2$

CASE "!"

h2$ = stack$(stackLevel)

checkParen h2$, "+-*/"

stack$(stackLevel) = h2$ + c$

CASE "r"

h2$ = stack$(stackLevel)

checkParen h2$, "+-*/"

stack$(stackLevel) = "sqrt(" + h2$ + ")"

END SELECT

END IF

IF stackLevel <> 1 THEN

REM

END IF

PRINT l$, stack$(1): PRINT #2, l$, stack$(1)

LOOP UNTIL EOF(1)

SUB checkParen (s$, ops$)

pLevel = 0

flag = 0

FOR i = 1 TO LEN(s$)

SELECT CASE MID$(s$, i, 1)

CASE "("

pLevel = pLevel + 1

CASE ")"

pLevel = pLevel - 1

CASE ELSE

IF pLevel = 0 AND INSTR(ops$, MID$(s$, i, 1)) THEN flag = 1: EXIT FOR

END SELECT

NEXT i

IF flag THEN s$ = "(" + s$ + ")"

END SUB

Edited on May 31, 2016, 7:33 am

Edited on May 31, 2016, 7:34 am

Posted by Charlie on 2016-05-31 07:31:18

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