FT. Sum and Double, Double and Sum

Rob Fathauer discusses the ins and outs of the mathematical toy business, and we ask: For which numbers is the sum of digits the same as the sum of digits of twice the number. For example:

The sum of the digits of 351 is 9 and the sum of the digits of 2 x 351 = 702 is also 9.

1) If a number has this property, can we always rearrange its digits and obtain another number with this property (513, 135, etc all have it)

2) Which powers of 2 have this property?

3) And most of all, can you give a simple characterization of the numbers with this property, in terms of just the digits themselves?

1. Brian Tristam Williams said,

May 23, 2009 at 5:59 am

I was going through powers of 2 after your last podcast. I couldn’t find any that had this property, and I went up pretty high.

However, I did spot an interesting pattern that yielded just one Google search result, which was the subject of an earlier podcast of yours. Sweet!

If you go through all the powers of 2, you will find that the very last digit ends in the following sequence 1 2486 2486 2486 …

Now for the second digit (the tens) – it ends in the following sequence: [One]362512499863748750[Zero] [One]362512499863748750[Zero] …

Now, doing a Google search for [One]362512499863748750[Zero] yields exactly one result. Nice! A Google ofÂ  “[One] 3 6 2 5 1 2 4 9 9 8 6 3 7 4 8 7 5 0 [Zero]” also yields another, different single result.

Of course, by posting this here, that property is likely to change, but it was fun while it lasted :-)

[[Edit by strauss: can’t spoil the fun! So messed with the digits to keep it out of google!]]

~podcast listener in Johannesburg, South Africa

2. Brian Tristam Williams said,

May 23, 2009 at 8:10 pm

Heh, you forgot to mess with the digits a little further up :-)

3. strauss said,

May 25, 2009 at 6:42 pm

Fixed! Hope I got to it before Google!