A square number is a number that can be written in a perfect square, that is  or . This hat contains those numbers and multiplies of them. But what other interesting things will we find in this hat.

When I first read up about Hat 0, I was astounded to have seen the little connections it has among the numbers. While I do not have the proof for some of these connections, I hope you simply marvel at them.

Mathematicians know that the probability that a number is not located in hat 0 tends towards  or 0.6079. Ain’t that something? The number  has appeared in a sequence with apparent link with trigonometry or geometry. Out of the first 100,000 numbers,  predicts 60,703 numbers not in Hat 0 or 39,207 numbers appearing in the hat. The actual figure is 60,704 but that is sure very close to what  predicts.

The next interesting thing about Hat 0 is a small little anomaly in another sequence we can define in the hat. Let’s see what is it.

We define another sequence where the terms involve the occurrences of consecution numbers. Looking back at our original sequence of Hat 0, the first occurrence of two consecutive numbers occurs at {8, 9}. Later, that of three consecutive numbers occurs at {48, 49 50}. We will do is that we will list the smallest term of the first run of n consecutive integers. That gives us

4;
8;
48;
242;
844;
22,020;
217,070;
1,092,747;
8,870,024;
221,167,422;
221,167,422;
47,255,689,915;
82,462,576,220;
1,043,460,553,364;…

If you would take a close look at this sequence of numbers, you would notice that the 10th and 11th term are the same namely, the smallest term of the first run of 10 and 11 consecutive integers is 221,167,422. It seems that this does not occur anywhere else in the sequence.

The strange thing about this is that I have not seen any other sequence, arithmetic, geometric, or otherwise, where identical consecutive terms only happen once.

How strange it seems is Hat 0.