My son is a poet:
There once was a number named ten
Who was round like an ‘O’ on one end
But his face was as flat
As his backside was fat
And the contrast drove him ’round the bend
There once was a number named ten
Who repeated again and again
It sure would be scary
If I were binary
I’d be eight parts less than I am
And yes, “ten” rhymes with “end” and “am”.
My son asks me “What do the numbers 26, 264, 307 and 836 all have in common?” After enjoying my puzzled look for a moment, he tells me all their squares are palindromes, but they’re not palindromes themselves.
Most palindrome squares are squares of palindromes, like (11 * 11) == 121, (121 * 121) == 14641, and (22 * 22) == 484. When the digits in palindrome number are small, so there is no carrying when you square, then the square is also a palindrome. But there are a few squares that are palindromes even with all the carrying. Like (307 * 307) == 94249.
Now I can’t let an 11 year-old show me up like that. I may not know all the squares to 1000, but I do know how to program, so I whipped up the following to find more of these squares.
# squares.py # Python code to find squares that are palindromes. __author__ = 'email@example.com' def is_palindrome( int_in_question) : """ True if int_in_question is an integer that is a palindrome when written in base 10. """ # Change the number into a string and then a list. as_list_of_chars = list( str( int_in_question)) # Copy the list and reverse it. reversed_list_of_chars = list( as_list_of_chars) reversed_list_of_chars.reverse( ) # True if the list of chars is palindromic. return as_list_of_chars == reversed_list_of_chars def print_palindrome_squares_up_to( limit) : """ Prints all the positive integers up to but not including limit, where the integer's square is a palindrome but the integer itself is not. Also prints the square. """ for i in range( limit) : if not is_palindrome( i) : if is_palindrome( i * i) : print( i, i * i)
Then I checked the first 100 million counting numbers. (This took a few minutes.)
>>> import squares >>> squares.print_palindrome_squares_up_to( 100000000) 26 676 264 69696 307 94249 836 698896 2285 5221225 2636 6948496 22865 522808225 24846 617323716 30693 942060249 798644 637832238736 1042151 1086078706801 1109111 1230127210321 1270869 1615108015161 2012748 4051154511504 2294675 5265533355625 3069307 9420645460249 11129361 123862676268321 12028229 144678292876441 12866669 165551171155561 30001253 900075181570009 64030648 4099923883299904 >>>
And there you have it. Integer sequence A059744.
Be sure to check out Feng Yuan’s palindrome page, which says 2,554,237,742,555,355,264,685,484 (more than 2 quindecillion) squared is palindrome 6524130445402999318864545454688139992045440314256.
During my son’s math lesson today we got on the subject of , which I prefer to write as . We made a table of a few obvious values and limits:
So the values rise from , peak somewhere in , and asymptotically drop to 1 after that. So the maximum is probably between 2 and 4, and I asked my son how we could find it. And he suggested we find where the curve is flat and the derivative is zero. And that’s when I realized I’d forgotten know how to find . My son is only 11 and we’ve only touched on calculus, so he couldn’t help. I figure the first step is to express it as , but I don’t know where to go from there. So we turn to the internet, and I read Wikipedia’s entry for the nth root algorithm, which doesn’t answer my question but is nice to read because it’s a little gem. And then we get to WolframAlpha, which tells us everything we want to know. See for yourself, the WolframAlpha page for , which says the derivative is , which is zero when or or . So the maximum of is . Not very surprising, but nice to have it confirmed. (When I described this problem to my neighbor he said the max would be at as soon as he heard it was between 2 and 4.) But I still don’t know how to calculate that derivative, except by asking WolframAlpha. Which brings up the question, is the internet making me stupid by making things easy? Or is it making me smarter by taking care of drudgery? I’d like to say it makes me smarter, but I know that, once in a while, what appears to be drudgery at first turns out to be important, providing unexpected insight.
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