All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Just Math
Consecutive Base Baffle (Posted on 2013-12-04) Difficulty: 3 of 5
Determine the smallest positive integer which is a palindrome in precisely 3 consecutive bases (excluding base 10). What are the next two smallest positive integers with this property?

*** Any solution must have more than one digit in any given base. So, trivial solutions like (0)base 3, (2)base 5 or, (C)base 14 are not allowed.

*** "Excluding base 10" means that the sequence of bases 9,11,12 for example, does not count as a possibility and, the bases have to be truly consecutive and all be on one side or the other of 10.

No Solution Yet Submitted by K Sengupta    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution computer solution | Comment 1 of 4

DEFDBL A-Z

OPEN "conbs.txt" FOR OUTPUT AS #2
FOR a = 1 TO 22
 FOR bs = a + 1 TO 23
  v = a * (bs + 1)
  PRINT #2, USING "############"; v;
  PRINT #2, USING " ##"; bs;
  PRINT #2, "  ";
  PRINT #2, USING " ##"; a; a
 NEXT bs
 FOR b = 0 TO 22
   IF b > a THEN st = b + 1:  ELSE st = a + 1
   FOR bs = st TO 23
    v = a * (bs * bs + 1) + b * (bs)
    PRINT #2, USING "############"; v;
    PRINT #2, USING " ##"; bs;
    PRINT #2, "  ";
    PRINT #2, USING " ##"; a; b; a
   NEXT bs
   FOR bs = st TO 23
    v = a * (bs * bs * bs + 1) + b * (bs * bs + bs)
    PRINT #2, USING "############"; v;
    PRINT #2, USING " ##"; bs;
    PRINT #2, "  ";
    PRINT #2, USING " ##"; a; b; b; a
   NEXT bs
   FOR c = 0 TO 22
     IF c + 1 > st THEN st2 = c + 1:  ELSE st2 = st
     FOR bs = st2 TO 23
      v = a * (bs * bs * bs * bs + 1) + b * (bs * bs * bs + bs) + c * (bs * bs)
      PRINT #2, USING "############"; v;
      PRINT #2, USING " ##"; bs;
      PRINT #2, "  ";
      PRINT #2, USING " ##"; a; b; c; b; a
     NEXT bs
     FOR bs = st2 TO 23
      v = a * (bs * bs * bs * bs * bs + 1) + b * (bs * bs * bs * bs + bs) + c * (bs * bs * bs + bs * bs)
      PRINT #2, USING "############"; v;
      PRINT #2, USING " ##"; bs;
      PRINT #2, "  ";
      PRINT #2, USING " ##"; a; b; c; c; b; a
     NEXT bs
   NEXT c
 NEXT b
NEXT a
CLOSE

SHELL "sort < conbs.txt > conbs2.txt"

OPEN "conbs2.txt" FOR INPUT AS #1
DO
  a$ = b$: b$ = c$
  LINE INPUT #1, c$
  IF LEFT$(a$, 12) = LEFT$(b$, 12) AND LEFT$(c$, 12) = LEFT$(b$, 12) THEN
    a = VAL(MID$(a$, 14, 2))
    b = VAL(MID$(b$, 14, 2))
    c = VAL(MID$(c$, 14, 2))
    IF b = a + 1 AND c = b + 1 THEN
       PRINT a$
       PRINT b$
       PRINT c$
       PRINT
    END IF
  END IF

LOOP UNTIL EOF(1)

finds

decimal base  representation
  178     6    4  5  4
  178     7    3  4  3
  178     8    2  6  2
  300     7    6  0  6
  300     8    4  5  4
  300     9    3  6  3
  373     8    5  6  5
  373     9    4  5  4
  373    10    3  7  3
  676    10    6  7  6
  676    11    5  6  5
  676    12    4  8  4
 1111    12    7  8  7
 1111    13    6  7  6
 1111    14    5  9  5
 1702    14    8  9  8
 1702    15    7  8  7
 1702    16    6 10  6
 2473    16    9 10  9
 2473    17    8  9  8
 2473    18    7 11  7
 3448    18   10 11 10
 3448    19    9 10  9
 3448    20    8 12  8
 4651    20   11 12 11
 4651    21   10 11 10
 4651    22    9 13  9
 


so the smallest such integer, in decimal, is 178, with base-6, 7 and 8 representations of  454, 343 and 262 respectively.

The next case is 300, where the base-7, 8 and 9 representations are 606, 454 and 363 respectively.

The next two cases involve base-10 and therefore are ruled out, so that the next sought case is decimal 1111, where base-12, 13 and 14 representations are 787, 676 and 595 respectively.

Further cases are also shown. Digits higher than 9 are shown coded as decimal, rather than the traditional A, B, etc.


  Posted by Charlie on 2013-12-04 17:56:33
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (8)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information