This is an attempt to explain what is going on without using any maths...
I'll get around to a mathematical description Real Soon Now!
-- Rob.
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OK. Imagine a lake with lots of stepping-stones (actually they are
points on an elliptic curve and there are 416363315556124458285894983
of them).
There is a sign on each stone indicating which stone to step to next
(actually there is a function, it doesn't really matter which one,
just that it is fairly random and fixed in advance.)
There are about 50 of us who each parachuted in to a random stone and
started following the signs, stepping from one stone to the next a few
tens or hundreds of thousands of times per second.
Every billion steps or so we find a stone which is "distinguished" in
some way, for instance it is particularly small, and before leaving it
we paint a mark on it indicating that we have been there (actually we
send some info about it to a central site by email).
After something like 30000 billion steps, one of us will step on a
stone that another of us has already visited. He'll continue
following the signs, just as the previous visitor did, and after a
while will find a stone that they painted a mark on.
Then we'll know that two of us have stepped on that stone and using
some information the two people recorded about the paths they took, we
can calculate the final answer.
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We are not sure exactly how many steps will be necessary because of
the randomisation but we can estimate how many.
The table below contains two columns: if at some stage we have marked
the number of stones indicated in the left column but have not found
the answer yet, then the expected number of stones left to mark before
we finish is given in right column.
Initially none were marked and we expected 30000 in total.
As I write this we have done 20000 (NB: there is an up-to-date total in
counter.text at the Web site). Reading from the table, we expect about
16056 more. At a rate of 1200 per day, this should take two weeks.
Note that 16056 is more than 30000 minus 20000 because we have to take
into account the fact that in theory we could have found the answer
already, but in fact that wasn't the case.
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Done : Expected remaining
==== ==================
0 : 30000 <- we started here on the 18th of December.
1000 : 29017
2000 : 28067
3000 : 27150
4000 : 26265
5000 : 25414
6000 : 24594
7000 : 23806
8000 : 23048
9000 : 22321
10000 : 21623 <- we were here on the 29th.
11000 : 20953
12000 : 20311
13000 : 19696
14000 : 19106
15000 : 18541
16000 : 18000
17000 : 17482
18000 : 16986
19000 : 16511
20000 : 16056 <- we reached here on the 6th of January.
21000 : 15620
22000 : 15202
23000 : 14802
24000 : 14419
25000 : 14051
26000 : 13699
27000 : 13361
28000 : 13037
29000 : 12725
30000 : 12426
31000 : 12139
32000 : 11863
33000 : 11598
34000 : 11343
35000 : 11098
36000 : 10861
37000 : 10634
38000 : 10415
39000 : 10204
40000 : 10000
41000 : 9804
42000 : 9614
43000 : 9431
44000 : 9254
45000 : 9083
46000 : 8918
47000 : 8758
48000 : 8604
49000 : 8454
50000 : 8310
51000 : 8169
52000 : 8033
53000 : 7902
54000 : 7774
55000 : 7650
56000 : 7530
57000 : 7413
58000 : 7299
59000 : 7189
60000 : 7082 <- hopefully we'll never get down here!
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