Yes... Another question is whether more than one amoeba can occupy a single position.
I imagine that this can happen, because there's no mention of divisions being blocked by crowding. But, it's worth verifying that assumption... Thanks, -- Raul On Sun, Sep 5, 2021 at 10:00 AM 'Mike Day' via Chat <[email protected]> wrote: > > Aaaaah.... > > Thanks all. I suppose the remark, that there are N+1 amoebas after N > divisions, was the clue, as well as your various examples. > > I’d thought they meant that all members of a generation spawn offspring; > evidently only one may reproduce at each stage. Onwards and downwards. > > Thanks again, > > Mike > > Sent from my iPad > > > On 5 Sep 2021, at 12:12, 'Michael Day' via Chat <[email protected]> wrote: > > > > The good news: 2 new problems yesterday after a lull for several months. > > The bad news (for me): I don't understand them! > > > > ... which is often the case, but here, I don't even get the Mickey Mouse > > case. > > > > It's not done to "spoil" these problems for others, but I don't think it's > > cheating to > > invite an explanation for how C(2) = 2 for problem 762. Once I understand > > what they > > want I can go on and (probably not) be able to solve it for myself. > > > > As I mis-understand it, the starting position is (0,0) and there's only one > > state at each generation. > > Where does the multiplicity arise? > > > > I should post my query in their Clarifications Forum, but don't fancy > > being trolled there - much > > better to be teased by fellow J(oker)s. > > > > The original of problem 762 can be found here: > > https://projecteuler.net/problem=762 > > > > Here's a copy, modified to non-graphics, roughly as pasted into my > > proto-script: > > > > ... better in a fixed width font: > > NB. Problem 762 > > NB. Consider a two dimensional grid of squares. The grid has 4 rows but > > infinitely many columns. > > > > NB. An amoeba in square (x, y) can divide itself into two amoebas to occupy > > the squares (x+1,y) and > > NB. (x+1,4|y+1), provided these squares are empty. > > > > NB. The following diagrams show two cases of an amoeba placed in square A > > of each grid. When it divides, it is > > NB. replaced with two amoebas, one at each of the squares marked with B: > > > > NB. (origin at J matrix index 3 0) > > NB. ('abb' (3 2 3, each 0 1 1) } 4 6 $ '.'),.(4 4$' '),. 'abb' (0 0 3, > > each 3 4 4) } 4 6 $ '.' > > NB. ...... ...ab. > > NB. ...... ...... > > NB. .b.... ...... > > NB. ab.... ....b. > > > > NB. Originally there is only one amoeba in the square (0, 0). After N > > divisions there will be N+1 amoebas > > NB. arranged in the grid. An arrangement may be reached in several > > different ways but it is only counted once. > > NB. Let C(N) be the number of different possible arrangements after N > > divisions. > > > > NB. For example, C(2) = 2, C(10) = 1301, C(20)=5895236 and the last nine > > digits of C(100) are 125923036. > > NB. Find C(100,000), enter the last nine digits as your answer. > > > > One for Joseph Turco, perhaps; and thanks for any, non-spoiler, tips! > > > > Mike > > > > -- > > This email has been checked for viruses by Avast antivirus software. > > https://www.avast.com/antivirus > > > > ---------------------------------------------------------------------- > > For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
