# How many lines are needed to connect all smiling toasters in a 4x4 grid?

The puzzle is as follows:

How many straight lines do you need to draw the least possible to join all the smiling toasters if you should not raise the pen or go over any line already drawn? Remember that it is allowed to cross.

The alternatives given are as follows:

1. 6
2. 5
3. 7
4. 4

How should this puzzle be approached? I'm getting 7 lines, however, I think there are different ways. Is there a way to minimize the trials?

I found this riddle in a book Logical Challenges from 2000's. It seems to be an adaptation from a reprinted copy of Martin Gardner's Puzzle's book from 1970s.

Because this puzzle has a drawing it would help if answers also included drawings so I could properly visualize the lines and understand why they are there.

• Related, related, related. I haven't found an exact duplicate yet but would be shocked if none existed. Feb 24, 2021 at 4:17
• I think there are different ways. - Yes. Is there a way to minimize the trials? - Find out what you can do before trying. In the case of this puzzle, the important parts are: you can draw to the outside of the region, and you can draw diagonals. After you realize them, it's just a matter of trying harder until your answer agrees with the book. Feb 24, 2021 at 4:35
• Optimization problems are nasty, and this is also a problem with your questions: we can tell you what you need to realize, and we can tell you what the answer is, but we cannot tell you why it is the answer, at least logically. Feb 24, 2021 at 4:38
• @Bubbler To your first comment, yes I did several trials and attempted what you mentioned but I still got stuck. I am sorry if I had given the impression of lack of logic. My intented question is to ask a justification how to minimize a certain feature. I know the nature of a puzzle is to be the one who solves it. Perhaps this part I'm not good at. Hence I need guidance. This makes me sad. Feb 24, 2021 at 5:22

Here is a solution with six lines:

It's difficult to tell how I found this, except that I already knew a solution for a 3 by 3 grid with 4 lines, which can be found e.g. here on our sister site Mathematics Stack Exchange. It's also possible that a solution with 5 lines exists.

(By the way, the puzzle is missing the requirement that the lines must be orthogonal or diagonal. Otherwise you can simply do something like this, if you extend the lines far enough upwards and downwards.)

• Or I've read somewhere that people get creative with "using a pen which tip width is as wide as the image, so using only one line". Feb 24, 2021 at 17:52
• @Glorfindel I'm accepting your answer. To be honest I had seen a similiar solution for a grid of 3 by 3. Is it okay to assume that if you increase the number of rows so all is a square the number of lines will be one unit less?. Assuming that the requirement is what you mentioned?. By the way, I agree that there is missing a requirement you stated. I was about to ask for the other solution and then I seen your link. That was very clear. I didn't know that the lines could cross through the toasters if they were far enough. Feb 25, 2021 at 2:01
• A cleaner additional requirements would be to require that the line pass through the center of the toaster. Feb 25, 2021 at 7:07

I don't know if this satisfies the intended problem but it follows the rules. I approached it by looking at the minimum multiple choice answer, and trying to exclude solutions of that size.

• I mentioned such a solution in my last paragraph – I think the rules of the puzzle as quoted are incomplete. Feb 24, 2021 at 10:30