Confuse Us, Oh, Leroy

Blog post # 79:

Over the last 4 or 5 days, these are the only two pictures I created even slightly worthy of posting here. And even they suck. Sorry.

Ascent Of Everythings

(Yes, I used the plural of "everything" in the name.)


Numbers Imagined Via Magnitudes


---------------------------

My last poll received a relatively large number of votes. (Hey, 9 votes IS a large number for one of MY polls!)
So, I will follow that poll with a poll guaranteed to get VERY FEW responses. What the heck.

Question:
What is (the 10th decimal digit after the decimal point of pi) times (the 10th prime) plus (the square of (the 10th composite))?

By the way, 1 is not considered a prime or a composite. And I'm talking about positive integers here.

Possible answers:

319
327
393
401
453
469

I will post the answer someday, but not until the poll closes.


------------------------------

Had enough of math? Too freaking bad! Be glad I'm not posting another poem today, at least!

Here is a puzzle.
I came up with this puzzle, but didn't solve it. Rob Pratt found a solution, though. So at least one solution exists. There may be many more.

Start with a 9-by-9 grid.
Place any number of black stones and whites stones on the squares of the grid so that each square has exactly one stone.
For a given row of the grid, take the lengths of (number of stones in) the runs of black stones and white stones and multiply these lengths.
(By "run", it is meant a string of consecutive stones in the row (or column) all of the same color, bounded by stones of the opposite color or by the edge of the row (or column).)
Do this for all rows and all columns to get 18 products.
Is it possible to place the stones so that all 18 products are distinct from each other?

For those of you who are confused about what I am asking, here is a sample 9-by-9 grid with just 16 distinct values that occur. (14 products don't occur elsewhere.)

1 = black, 0 = white (although it could be the other way around just
as easily).
1 0 1 0 0 0 1 1 0
1 0 0 1 1 1 1 1 0
0 1 1 1 0 0 0 0 0
1 0 0 1 1 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 1 1 0 0 0 1
0 0 0 0 0 0 0 1 1
0 0 0 1 1 0 0 1 1
1 1 1 0 0 0 1 1 1
The row products are:
1*1*1*3*2*1 = 6, 1*2*5*1 = 10, 1*3*5 = 15, continuing:
16, 9, 18, 14, 24, 27.
And the column products:
2*1*1*4*1 = 8, 2*1*5*1 = 10, 1*1*1*5*1 = 5, continuing:
3, 1, 7, 12, 24, 20
So, the products 2 and 4 don't occur in my partial "solution".
While 10 and 24 each occur twice. Not acceptable.

I will not give a solution myself. But if any of you find a solution, please post a comment with the solution you found to this blog-post.


Update: (Jun 12, 5:17 am) Richard Heathfield has found several solutions as of now, and Rob Pratt has found another solution.

---------------------------


And finally, my webpage of computer art has been updated!

MANY pictures there.

www.
[Update: URL censored because site has been hacked!]
Check it out.

---------------------------

Leroy