Problem 762

[KING-OLE]

I don't understand the "number of different possible arrangements after N divisions".

If you have 4 rows, and you place the A in each row, the outcome will have 4 different arrangement regardless of how many divisions follow.

For C(2), I get 4 arrangements if first amoeba is placed in first column (see drawing).

What am I not getting?

[jaap]

Two things:

Originally there is only one amoeba in the square (0,0).

So the first amoeba is in the bottom-left corner.

After N divisions there will be N+1 amoebas arranged in the grid.

A division is a division of a single amoeba, not the division of an entire generation. When an amoebe splits, it disappears leaving two new amoebas in the next column, so each division increases the number of amoebas by one. So for N=2 there will be three amoebas at the end, one in column 1 and two in column 2. There is no fixed order in which the amoebas split, and some may never split.

[KING-OLE]

Oh, ok thanks - so if I understand you correctly, these would be the 2 arrangements after C(2)...
.

p762.png

[neverforget]

Yes, though keep in mind that the grid continues to the right (only the height is constrained to 4 cells) and the individual amoebas are indistinguishable.