M,

Here is what you are asking for:

"1111" has other 9 equivalent combinations, they are:(2222, 3333, 4444, ......, 9999) all equiprobable

While "1112" has much more equivalent combinations:(1113, 1114, ......, 1119, 2221, 2223, ......, 2229, ......, 9998) all equiprobable too

I think this is where you making an error, M. There aren't more equivalent combinations of 1112.

On the other hand, I believe you're correct in saying that 1111 is somehow different to a number such as 3071. They certainly look different to me. So how are they different?

To make it clear, let me first reduce the number of digits in your example, so it reads:

"111" has other 9 equivalent combinations, they are:(222, 333, 444, ......, 999) all equiprobable

While "112" has much more equivalent combinations:(113, 114, ......, 119, 221, 223, ......, 229, ......, 998) all equiprobable too

OK? The sense remains the same.

Now, let's clarify the algorithm used to generate the numbers in your first series:

*1. For the first digit, select any number from 0 to 9.*

2. For the second digit, add 0 to the first digit.

3. For the the third digit, add 0 to the first digit.(For short, let's name this algorithm

*Add 0, Add 0*).

As you say, that generates 10 numbers, 000, 111, 222, etc.

Obviously, we need a hundred of these algorithms to generate all 1000 numbers.

Your next series starts 112, which uses algorithm

*Add 0 Add 1*.

That generates 112, 223, 334 etc... (ten numbers only, just as in the first series).

It doesn't generate, as you claim, 221, or 229, or 998, so those are not equivalent combinations - they are generated by

*Add 0 Add 9, Add 0 Add 7* and

*Add 0 Add 9* respectively.

So the question remains, why does 111 seem different to 496, if they're both equally probable?

Well, if the structure of the algorithm is

*Add x, Add y*, any algorithm that has a value of zero for x or y immediate becomes noticeable to our pattern-sensitive perception, because it makes a pair with the first digit.

eg 484 661 585 - we see the pattern immediately.

Similarly, any algorithm in which x=y also becomes noticeable for the same reason, that it creates a pair. 388 611

If x doesn't equal y, the pattern doesn't stand out (though it's still there). 493 217

(Now I'm halfway through this, I've just realized what M was trying to say in his quote above - but I'll plough on anyway).

But, of the 100 available algorithms, one stands out in our perception because it forms not just a pair, but a triple, and triples are more noticeable than pairs.

*Add 0 Add 0* is special for this reason, and the numbers that it generates stick out like a sore thumb.

Likewise,

*Add 1, Add 2* sticks out, as it generates the familiar 123 etc.

So M was throwing 229 and 221 into the same category, in the sense that they both contain a pair, and pairs are more common than triples.

Gnu.