Can you think outside the box to solve this problem?

inside the box:
4 apples (wrong, unless I gave the other 8 away – that would be outside the box)
8 apples (right, but I’d eat one while I’m picking which 8 I take, then 7 or 8 depending on if I decide to replace the one I ate)

a little outside the box:
12 apples, in two groups

way outside the box:
applesauce (I have no “apples”), because that’s why I bought all those apples in the first place.

z=y=x=15

I always pick up Games magazine, and I love doing the Cryptic Crosswords, which require out of the box thinking.
A Cryptic Crossword is probably the most difficult puzzle you can find. It has a clue, which tells you how long the word is, gives you a clue to what the word is, and also a second clue to help you create the word, should you be able to figure it out. The aspect that makes this difficult is that they don’t tell you what the clue is, per se.
For example: the clue is Ms. Derek returns to 5 and dime United States Patent. (7)
The answer has 7 letters. Ms. Derek is Bo. Returns might mean Bo becomes Ob.
5 in roman numberals is V. A dime is 10 cents. United States is US.
So the answer is Ob V 10 US or Obvious, which is a definition of Patent.
A puzzle like this can take weeks, and requires out of the box thinking.

@Ame_Evil nailed it.

2x+5y+z=120
3x+2y+3z=120
1x+6y+z=120

Can also be written as:

1x+6y+1z=120
2x+5y+1z=120
3x+2y+3z=120

8 is the common divider (1+6+1, 2+5+1, 3+2+3 all equal 8). From there on out it’s pretty easy to figure out what x,y and z are.

That is the reasoning that I was looking for.

To elaborate:
First notice that the sum of the coefficients is 8 in each case.
Then realize that this implies that all 3 variables can be given the same value.
Finally divide 120 by 8 to get the value for x, y and z.

@markyy Hmm I used some “in-the-box” thinking from my maths lessons 3 years ago.

1) 2x+5y+z=120
2) 3x+2y+3z=120
3) x+6y+z=120

2x+5y+z=x+6y+z (because they both equal 120 therefore 1 and 3 equal each other)

x-y=0 (simplify)
x=y

7x+z=120 (change y terms to x in either 1 or 3)
5x+3z=120 (change y terms to x in 2)
21x+3z=360 (multiply 1 or 3 to get 3z)
16x=240 (subtract 5x+3z=120 from 21x+3z=360)
x=15 (solve for x)

15*7+z=120 (substitute x=15)
105+z=120
z=15

therefore x=y=z=15

I like to think inside the box.

Btw on an interesting note, did you know that the term “thinking outside the box” refers to this classic problem:

http://richardwiseman.files.wordpress.com/2009/02/nine-dots-puzzle.jpg

Where you have to connect all of the nine dots with only drawing four straight lines.

What not many people know is that you can also solve this by only using three lines as well. Have a go thinking out of the box in this one :D. Literally.

I don’t think out of the box thinking is necessary to solve this. But for unifying general relativity and quantum mechanics it is. Or how to accommodate 9 billion on our planet in 2050.

Matt, there are different degrees of out of the box thinking. If I were able to think way outside of the box I would have had a theorem attached to my name, or at least a lemma.

I’m horrible at math so my first thought was ABC’s and 123’s. That alphabet/numerical soup from my childhood.

@Ame_Evil I’ve got the four but three??

edit!: got it!! lol

No way that it can be done with 3 lines, at least the way I remember the problem as being stated. You have to be able to do it without lifting the pencil or retracing. In order to be able to do that with 3 lines it would be necessary to average an additional 3 points per line. Unless you are working in some space where the parallel lines intersect, it can’t be done.

I’m eating Cheerios out of the box – does that count?

oh maybe I’ve got it totally wrong then… if you can’t lift your pencil…that’s pretty impossible unless you’re doing paper gymnastics???

oh and…@Zen_Again…priceless!! I love this place!

@LostInParadise I’ll give you a small (but large) clue how to solve it before giving you the solution. You don’t need to go through the centre of each dot.

You can also do it in one line. With a big enough pencil :D.

I can see if that if you make the dots wide enough or, equivalently move them sufficiently close together, you can do it with three lines, but in the diagram as you showed it, I don’t see how it can be done.