Wednesday 30 May 2012

The Cubic Attractiveness of 153

Just in time for June 1st, which is the 153rd day of this leap year.  I recently discovered an interesting quality about the number 153.

Ok, I was amazed. I was lead from a comment (below) that this is all shown in the online integer sequence, but it is still great fun.

Pick any old number you want, and multiply by three (or just pick a number that is a multiple of three).
Now take all the digits and cube them and add the cubes together.
For example, if you picked 231, you would add 23 + 3 3 + 1 3 to get 36. 
Yeah, So what you're probably thinking... but take that new number and do the same thing... cube the digits and add them up... Nothing?  Keep going... eventually you get to 153, and then when you  do it again, you get 153 forever.

In slightly more formal language, it seems that 153 is the fixed point attractor of any multiple of three under the process of summing the cubes of the digits.

The 231 that I used above goes to 36, which goes to 243, then 99, 1458, 702 and 351, which give 153. 

If you started with something like 72, it gets there pretty quickly,  72 →   351 →  153.  

Other numbers take a little longer.  717, for example, takes 13 iterations of the cubing process to get there...but it does.  
And 153 is special... ONLY the multiples of three go there.  Pick another number that's not a multiple of three and run the process and several things might happen, but what won't happen, is going to 153. 

When I began to explore why this happened I could figure out a few things easy enough.  The first realization was that the iterations couldn't just get bigger and bigger and diverge to infinity.  93 is 729 and so any four digit number has to iterate to a value less than 4 * 729 or 2916, and anything with more digits just keeps getting pushed down until it gets under that limit.
So all the numbers in the world have to end up doing something other than just keep getting bigger.
That only leaves a few options.  They might just keep jumping around in some orbit that cycles through several numbers.  This happens with 46 for example. 
After a few iterations you fall into a three cycle. 

46 → 280 → 520 →  133 → 55  → 250 →133

Other numbers also do this, and there are a few different three cycles and  two cycles, but that is sort of an oddity, at least for numbers less than 1000 (still have work to do here).
Most numbers go to a fixed attractor.  For multiples of three, that seems to always be 153.   For numbers that are not multiples of three, the general fixed points are either 370, or 371.  And they have a modulo three relationship as well.  Numbers that are congruent to 2 Mod3 (they have a remainder of 2 when you divide by three) generally go to 371.  The exceptions are a couple of numbers; 47 , 74, 77,  89 & 98,   which go to 407 (of course 707, 908, 980, etc would also, I counted 30 numbers less than 1000 that go to 407 as a fixed point).  
All the cycles that I have found are numbers that are equivalent to 1 Mod3.  The cycles seem pretty common with less than 1/2 the smaller numbers going to a fixed value of 370 and the occasional few like 1, 10, 100, that have a fixed point at one.  118 is in that group as well, for example.  

The reason for the separation into modulus classes of three is easy enough to explain.  When you cube a number, it's modulus in base three isn't changed.  For example, 4 Mod3 is 1, and 43 = 64 is also equivalent to 1 Mod3.  So the Modulus of a number doesn't change under this process, grouping the results together.  

Now when you add in the fact that there are not really that many of these smaller (less than 3000, say) numbers that can be made with the sum of cubes unless you allow for weird numbers like 11,111 or something to get five.  
I haven't explored much beyond 1000, so I'm not sure if I will come across other cyclic orbits. Or why only the 3n+1 type numbers produce cycles. And I'm not sure what else I may find, but I'm thinking that what I've seen so far makes 153 a very special number. 


Sue VanHattum said...

Pat, this is cool. I think this post is a great reply to anyone who thinks there's nothing new to discover. (May I include it in the next Math Teachers at Play?)

timteachesmath said...

What a fun problem. I'm still looking around, but I found mention of it at

Tim said...

You got 'em all.

Unknown said...

I was perusing through your blog and I found this. This is really interesting, I like this a lot. I decided to do a similar experiment; however, I squared the digits instead of cubed them. It turns out, MOST digits eventually go to 4 and then do a loop (4 -> 16 -> 37 -> 58 -> 89 -> 145 -> 42 -> 20 -> 4 ...). However, there are some numbers that go to 1. Unfortunately, I'm not able to find a pattern with these numbers. For all one and two digit numbers, I found the numbers 1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, and 97 tend to the number 1. Like I said, I couldn't find a pattern with these. But, I believe I can say the following:

If you have any multiple of three, and you keep squaring the digits and adding them, you will eventually get to 4 and repeat a 9-number pattern.

Just an interesting find...but I'd like to find a pattern with the ones that converge to 1.

Pat's Blog said...

Derek, HAPPY to point out that you found the idea behind "Happy Numbers." I only heard of it awhile back when I had the same thought. Here is what I found

Unknown said...

Now, I'm figuring it out if you take the fourth power of these digits... One step forward! And, I'm finding some interesting results. There seems to be a fixed number here but it might be for all the numbers. I've found that they all eventually go to 13139 then it goes through a loop (13139 --> 6725 --> 4338 --> 4514 --> 1138 --> 4179 --> 9219 --> 13139...)
Now, I have only completed numbers 2 through 9 inclusive (it's nearly 1 AM and I'm getting tired!). Now, 1, 10, 100, etc. will obviously stay at 1. But for all the non-multiples of 10, it might be 13139 for all of them. Perhaps a good blog for your audience to check out.

Unknown said...

Contradiction!! No! For 12, I get 12 --> 17 --> 2402 --> 288 --> 8208 --> 8208 --> 8208 ... So, I guess we do have some contradictions (288, 828, 882, 71, 242, 224, 422, 21, etc.) and probably more. Well, maybe it's still worth a blog post. Maybe if it goes to 8208, it's a double happy prime (since it's the square of a square)!

Numbers are real said...

Hi, Pat. I know this is an old post, but if you see this, take a minute to read Ch. 21 in the Gospel of John--maybe the most Pythagorean chapter in the bible?