Is it possible??

Nappy

Milkcrate

Ok, you have a square crate holding 36(6 x 6) bottles. Can you arrange 14 bottles in the crate so that there is an even number of bottles in every row and column???

Thoie

First thought is no, because 14 is only divisible by 2 and 7, and 7 is an odd number.

Doodling on some paper shows that we can have
2 columns (height 7 - wrong)
3 columns - wrong
4 columns - heights 3 and 2 - wrong
6 columns - heights 3 and 2 - wrong


So I'd still say no.

Pherekydes

Yep, it can be done.

Hint: What six even numbers add up to 14?

red_fox

It's important to remember that you can place them anywhere in the crate, you don't have to stack them neatly

Thoie

red_fox wrote: »

It's important to remember that you can place them anywhere in the crate, you don't have to stack them neatly

*OCD Girl's head explodes*

Not stack them neatly! What kind of slovenly organisation is this!!!

And OK, thanks Pherekydes, think I've got it now That's not very neat though :P

Pherekydes

Can you do it with an odd number of bottles in each row and each column?

You must still use 14 bottles.

Nappy

Can someone somehow draw this, i cant get it??? How can you possibly have an even number of bottles in each column AND row??

Pherekydes

Howzat?

Nappy

Jeez!! Thanks, how dya go about workin that out? Any method or dya just play round 4 a while until ya solved it?

ZorbaTehZ

But the real question is... why would you keep your milk in conical flasks?

Pherekydes

Just sketched it out. It wasn't a tough one to work out.

Thoie

As requested above, is there a way to work this out without messing around with drawings?

Just in case I decide to start putting an infinite number of conical milk bottles into an infinitely large milk crate in an untidy fashion. What? Sometimes I get bored!

red_fox

Thoie wrote: »

As requested above, is there a way to work this out without messing around with drawings?

Just in case I decide to start putting an infinite number of conical milk bottles into an infinitely large milk crate in an untidy fashion. What? Sometimes I get bored!

Well if you want to fit a number of bottles which is divisible by four then neat stacking will work, if not (but still even) then you need the top right 4x4 of the diagram and fit the rest in as blocks of four (2x2) anywhere else.

Another question would be to ask what is the smallest crate you can fit 14 bottles in such that it satisfies the condition, and then for any even number.

As for the 6x6 with odd numbers in each row/column, I imagine it would be an easier problem as you could have a row/column with only one, and example of 14 into 6x6, where 0 is a bottle and - is empty would be:

0

---

00-0--
000---
000---
000---
-0----

Edit: if you have fewer then 10 bottles then only even numbers not divisible by four are 2 and 6, 2 is not possible and 6 would work something like:

0-0
-00
00-

which would give you a different solution to the 14 into 6x6 case

0-000-
-0000-
00-00-
---00-

---

---

fitting it into a 4x5 crate.


edit2: oops, an odd number in each row/column would disallow empty rows/colums so something like:

0---00
-0--00
--0---
---0--
00--0-
00---0

ShayK1

busy day eh?
Here's my solution...