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?
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
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Posted by Charlie
on 2016-06-01 08:04:18 |
Part 1 redone allowing unary negation
