||-|||--|-
-||--|-|||
|-|-|||--|
||||--|-|-

------|||||
----||---||
--||-|--|||
||-||--||-|

What is the next row of each sequence?

It might be helpful to change each - to a 0 and each | to a 1.

The first sequence becomes:

1101110010
0110010111
1010111001
1111001010

where each bit is the binary sum of the two bits above it and to the right (with a carryover of the leftmost bit to be used in calculating the rightmost).
In binary addition, 0+0=0, 0+1=1, 1+0=1, and 1+1=0. So, the result will be 0 if the bits above it are the same, and 1 if they are different, as someone noted earlier.
So, this isn't exactly a binary addition, but a twisted variation of a bitwise addition.
If this is the case, then the next few rows are:

0001011111
0011100001
0100100011

However, if that is the case, then the first row must have been selected arbitrarily, so it's possible that something else may have been intended altogether.
1101110010 in binary becomes 652 in decimal; I don't see anything significant about that number. The other three rows are binary representations of 407, 697, and 970, respectively. Nothing interesting there, either.
Perhaps we've solved it; perhaps there is something more to consider.

The second pattern, when changed, looks like:

00000011111
00001100011
00110100111
11011001101

A few general observations can be made about this series; each row has first 6, then 4, then 2, then no zeroes at the beginning. Then come at least two ones, with some intermittent zeroes and ones, and at least one 1 at the end. Perhaps, if each row depends on the previous one, the ones propagate down from row to row. How this may work is not immediately apparent, so I'll just throw that idea out there.
The binary rows, when represented in decimal, are 31, 99, 423, and 1741. I don't see much relating these numbers, except for the fact that they are all odd (a result of the last bit being a 1).
Any other ideas?

Posted by DJ on 2003-10-05 20:50:27