Math Games #1

Back at the start of March Mathness I promised the math would be “fun” (really!), but anyone would be forgiven for thinking the previous two posts about Special Relativity weren’t all that much “fun.” (I really enjoy stuff like that, so it’s fun for me, but there’s no question it’s not everyone’s cup of tea.)

Trying to reach for something a bit lighter and potentially more appealing as the promised “fun,” I present, for your dining and dancing pleasure, a trio of number games that anyone can play and which might just tug at the corners of your enjoyment.

We can start with 277777788888899 (and why it’s special).

Before I start, credit where it belongs, these all come from the YouTube channel, Numberphile, which is a great place if you’re fascinated by numbers.

(I’ve mentioned them many times before. They’re a favorite channel.)

Speaking of which, here’s their video about 277777788888899 (and why it’s special):

The topic here is multiplicative persistence, which is how many iterations of multiplying all a number’s digits together are possible until the result is a single digit.

For one example, the year 2019 has a persistence of zero, because:

No steps are possible. The zero in the original number insures that. (And any result with a zero will likewise end the iterations.) Likewise, any number with a five will also end the sequence pretty quickly.

As another example, let’s consider a shorter version of the “special number” (which we’ll get to in a moment), 2789:

Which is a persistence of just one.

What makes this game so interesting is that persistence turns out to be much shorter than one might think going in. Most numbers terminate surprisingly quickly.

In fact, of the integers less than 100, only 77 has as many as four steps:

What makes 277777788888899 special is that it has a persistence of 11, which is the highest known persistence.

Further, it is believed 11 may be the highest persistence of any number — nothing higher has been found so far.

Note that there are infinitely many numbers with a persistence of 11 (and all the shorter ones). The “special” ones are the ones with the lowest value for a given persistence. (Ten, for instance, is the lowest number with a persistence of one, but infinitely many higher numbers have the same persistence.)

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Next up, the Collatz Conjecture!

It’s an idea almost as simple as multiplying digits together, and it shares the goal of “reducing” an input number to, in this case, the number one (rather than any single digit).

The conjecture is that all positive integers do reduce to one.

Here’s a Numberphile video that gets into it (visually colorfully and very much worth watching):

The idea is that we recursively apply one of two possible rules to the input number, which results in a new number that restarts the process.

The rules are, for some number, N:

1. If N is even, calculate N÷2.
2. Otherwise, calculate (N×3)+1.

That’s it, that’s all there is to it. Take the result (if it’s not one) and keep repeating until you get one.

The conjecture conjects you always will.

The fun part is how the sequence can jump around until it (inevitably) reaches one.

As an example, let’s consider 77, which had the highest multiplicative persistence (4) of the numbers less than 100. Its Collatz sequence is:

77, 232, 116, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1

So 77 has 23 steps to reduce to one and rises as high as 232 along the way. For something a bit more impressive, consider 327:

327, 982, 491, 1474, 737, 2212, 1106, 553, 1660, 830, 415, 1246, 623, 1870, 935, 2806, 1403, 4210, 2105, 6316, 3158, 1579, 4738, 2369, 7108, 3554, 1777, 5332, 2666, 1333, 4000, 2000, 1000, 500, 250, 125, 376, 188, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1

Which has 144 steps and gets as high as 9232! It’s the winner among numbers up to 500.

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As the video gets into, because all sequences lead to one, we can generate a tree structure where each node and leaf is a number.

The resulting tree is an extremely sparse binary tree, because any given number node can only have two dependents, one representing each of the two rules.

The node’s parent is the result of applying the rules to the node, there is only one possible result, so only one possible parent.

But while two child numbers can have the same parent (one by dividing by two, the other with the multiplication+addition), it is far more common that there is only one valid child.

The resulting tree, especially if rendered as shown in the video, is strikingly complex for such a simple process.

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That simple processes can lead to complex outcomes has been demonstrated repeatedly in mathematics and physics. It seems to be a feature of reality.

The stunningly beautiful Mandelbrot is, at heart, the result of nothing more than repeating:

For each C (coordinate) until the new Z goes above 2.0 (or you get tired of doing it because it didn’t).

And we can be thankful for this potential to create complex systems, because otherwise we wouldn’t be here!

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Okay, one last bit of “fun” — the Dollar Game (how can it not be fun, it’s about money):

Full disclosure: I included this one to fill out the trio (I wanted three examples for the next bit). I haven’t explored it like I have the first two.

As such, I’m not going to get into this one, but I do want to point out a common aspect of all three of these, and in general of most of these mathematics videos:

Note the love, excitement, and enthusiasm, these mathematicians show for their craft. They just light up with delight!

Speaking from personal experience, it is a bit like a narcotic. Or a roller coaster — exhilarating!

I suspect part of it involves that math may be the one place in life where it’s possible to have absolute truth. Nothing is as real as mathematical truth.

There is also something really cool about how numbers work. The allure can be almost like that of a campfire or sea waves or clouds. Endless variation and complexity.

It’s very easy to see why physicists and mathematicians can become caught up chasing the elegance and beauty of dancing numbers (cough — String Theory — cough).

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All this delightful “fun” aside, there is a certain commonality among all three of the number games here. Generally it’s true of any number game.

It’s that these are all algorithms, all computations.

There is a series of steps performed, where (crucially) later steps depend on the outcome of earlier steps.

This differs from an evaluation where all inputs are available to combine to form an output.

For instance, the fraction 4/5 can be seen as a long division requiring multiple steps — an algorithm — or as the value that represents (without any implied processing steps). In fact, the latter is most correct in this case.

Even something like pi is a specific value despite that we have no way to name that value other than through a series of steps, an algorithm.

But there is no way to find the multiplicative persistence, or the Collatz sequence, without performing the necessary steps. There is no (single) mathematical expression that provides those values.

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I mention this only because not everyone makes the distinction between evaluation of a quantity and algorithmic computation.

It’s true that, in terms of a result, there isn’t much difference (as the example of 4/5 shows).

In the context of computer science, it is a crucial distinction tied to what we mean by computation and, more importantly, the limits of what can be computed — that’s where it becomes significant.

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But mostly these caught my eye and seemed like fun. Maybe you found them interesting, too.

Me? I totally get that light in their eyes!

Stay computational, my friends!