perplexus.info :: Numbers : Maximum Palindrome Count Muse

Maximum Palindrome Count Muse (Posted on 2022-06-07)

A 3-digit decimal (base ten) Palindrome P is such that when expressed in base N, with N being a positive integer between 2 and 36 inclusively (base 10 excluded), at least one of them is also a palindrome.

M denotes the total count of such non base ten palindrome for a given value of decimal palindrome P.

Which value or values of P maximises the value of M?

No Solution Yet Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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I get:

Pmax is 252

Mmax is 5

252 yields 5 palindromes when changed to other bases as follows:

(base, palindrome): {(5, 2002), (17, EE), (20, CC), (27, 99), (35, 77)}

in addition to base 10 (10, 252)

Here is the list of 3 digit palindromes and the number of "other base" palindromes for each:

101:0,

111:2,

121:3,

131:0,

141:1,

151:1,

161:1,

171:2,

181:1,

191:2,

202:0,

212:1,

222:2,

232:2,

242:3,

252:5, <---- winner, winner, chicken dinner

262:0,

272:1,

282:2,

292:2,

303:0,

313:2,

323:2,

333:2,

343:3,

353:2,

363:1,

373:3,

383:0,

393:1,

404:0,

414:2,

424:0,

434:2,

444:2,

454:1,

464:3,

474:0,

484:2,

494:2,

505:2,

515:0,

525:2,

535:0,

545:1,

555:3,

565:1,

575:2,

585:2,

595:3,

606:0,

616:2,

626:3,

636:0,

646:2,

656:2,

666:4,

676:4,

686:1,

696:1,

707:0,

717:2,

727:1,

737:2,

747:0,

757:3,

767:1,

777:2,

787:3,

797:2,

808:0,

818:1,

828:2,

838:2,

848:1,

858:3,

868:2,

878:0,

888:2,

898:2,

909:1,

919:2,

929:1,

939:1,

949:1,

959:0,

969:0,

979:2,

989:3,

999:2

whichPals(252)

Outputs: [[5, '2002'], [17, 'EE'], [20, 'CC'], [27, '99'], [35, '77']]

----------

def base2base(n,a,b):

""" input n which is a string of the number to be changed

'a' is an integer representing the current base

to be changed to base b

"""

def dec2base(i,base):

""" INPUT integer in base 10, return string

of Base base equivalent. """

convertString = '0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ'

if i < base:

return convertString[i]

else:

return dec2base(i//base,base) + convertString[i%base]

if a == 10:

return dec2base(int(n),b)

if b == 10:

return int(str(n),a)

elif b != 10:

return base2base(int(str(n),a),10,b)

def isPalindrome(n):

""" return True if integer or string n is a palindrome """

n = str(n)

if n == n[::-1]:

return True

else:

return False

def npals(p):

if not isPalindrome(p):

return 0

m = 0

for base in range(2,37):

if base == 10:

continue

x = base2base(p,10,base)

if isPalindrome(x):

m += 1

return m

def whichPals(p):

if not isPalindrome(p):

return 0

m = 0

answer = []

for base in range(2,37):

if base == 10:

continue

x = base2base(p,10,base)

if isPalindrome(x):

answer.append([base, x])

m += 1

return answer

ans = {} # generate a dictionary shown above

for i in range(100,1000):

if not isPalindrome(i):

continue

ans[i] = npals(i)

Posted by Larry on 2022-06-07 10:06:07

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