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Beginning and End (Posted on 2015-07-11) Difficulty: 3 of 5
When the digit is placed at the beginning and end of a positive integer N, the new number is 99*N.

Find the smallest value of N.

No Solution Yet Submitted by K Sengupta    
Rating: 3.0000 (1 votes)

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The Fibonacci connection Comment 6 of 6 |
(In reply to More precision with UBASIC computer program by Charlie)

Noticing the first 5 digits of N being the first 5 Fibonacci numbers, one sees that the next one is 1 higher than the next Fibonacci number only because it is receiving a "carry" from the next digit over.


The below table shows the result of keeping track of a number, each successive value of which is the preceding value multiplied by 10 and to which the next Fibonacci number has been added after the multiplication by 10. Up to the last few digits they track the digits of N, either for the 22-digit case with appended digit 1 or the 66-digit case with appended digit 1:

 1 	 11 
 2 112 
 3 1123 
 5 11235 
 8 112358 
 13 1123593 
 21 11235951 
 34 112359544 
 55 1123595495 
 89 11235955039 
 144 112359550534 
 233 1123595505573 
 377 11235955056107 
 610 112359550561680 
 987 1123595505617787 
 1597 11235955056179467 
 2584 112359550561797254 
 4181 1123595505617976721 
 6765 11235955056179773975 
 10946 112359550561797750696 
 17711 1123595505617977524671 
 28657 11235955056179775275367 
 46368 112359550561797752800038 
 75025 1123595505617977528075405 
 121393 11235955056179775280875443 
 196418 112359550561797752808950848 
 317811 1123595505617977528089826291 
 514229 11235955056179775280898777139 
 832040 112359550561797752808988603430 
 1346269 1123595505617977528089887380569 
 2178309 11235955056179775280898875983999 
 3524578 112359550561797752808988763364568 
 5702887 1123595505617977528089887639348567 
 9227465 11235955056179775280898876402713135 
 14930352 112359550561797752808988764042061702 
 24157817 1123595505617977528089887640444774837 
 39088169 11235955056179775280898876404486836539 
 63245986 112359550561797752808988764044931611376 
 102334155 1123595505617977528089887640449418447915 
 165580141 11235955056179775280898876404494350059291 
 267914296 112359550561797752808988764044943768507206 
 433494437 1123595505617977528089887640449438118566497 
 701408733 11235955056179775280898876404494381887073703 
 1134903170 112359550561797752808988764044943820005640200 
 1836311903 1123595505617977528089887640449438201892713903 
 2971215073 11235955056179775280898876404494382021898354103 
 4807526976 112359550561797752808988764044943820223791068006 
 7778742049 1123595505617977528089887640449438202245689422109 
 12586269025 11235955056179775280898876404494382022469480490115 
 20365011074 112359550561797752808988764044943820224715169912224 
 32951280099 1123595505617977528089887640449438202247184650402339 
 53316291173 11235955056179775280898876404494382022471899820314563 
 86267571272 112359550561797752808988764044943820224719084470716902 
 139583862445 1123595505617977528089887640449438202247190984291031465 
 225851433717 11235955056179775280898876404494382022471910068761748367 
 365435296162 112359550561797752808988764044943820224719101053052779832 
 591286729879 1123595505617977528089887640449438202247191011121814528199 
 956722026041 11235955056179775280898876404494382022471910112174867308031 
 1548008755920 112359550561797752808988764044943820224719101123296681836230 
 2504730781961 1123595505617977528089887640449438202247191011235471549144261 
 4052739537881 11235955056179775280898876404494382022471910112358768230980491 
 6557470319842 112359550561797752808988764044943820224719101123594239780124752 
 10610209857723 1123595505617977528089887640449438202247191011235953008011105243 
 17167680177565 11235955056179775280898876404494382022471910112359547247791229995 
 27777890035288 112359550561797752808988764044943820224719101123595500255802335238 
 44945570212853 1123595505617977528089887640449438202247191011235955047503593565233 
 72723460248141 11235955056179775280898876404494382022471910112359550547759395900471 
 117669030460994 112359550561797752808988764044943820224719101123595505595262989465704 
 190392490709135 1123595505617977528089887640449438202247191011235955056143022385366175 
    1   kill "fibnos.txt"
    5   open "fibnos.txt" for output as #2
   10   Fiba=1:Fibb=1:N=1
   20   for Lvl=1 to 100
   30       H=Fiba:Fiba=Fibb:Fibb=Fibb+H
   40       N=N*10+Fiba
   50       print Fiba,N:print #2,Fiba,N
   60   next
   70   close #2

The output list was truncated so as not to widen the area of the comment.

  Posted by Charlie on 2015-07-12 08:52:04
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