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

 Consecutive Base Baffle (Posted on 2013-12-04)
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.)
 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

 Search: Search body:
Forums (0)