Math Is Fun Forum

Hi chen.aavaz;

I found an answer, which seems to be the only one for whole cents.

Did it by trial and error in Excel after setting up the relationships.

Is this just any old puzzle (in which case I can post the answer), or homework (for which you'd need a solution method and workings)?

Sorry, but I don't know how you'd go about it 'properly'. Someone else might, though.

Just as well that I didn't post the cryptic reply to Bobby's post #10 that I had intended to do right after his!

Hi MathsIsFun;

The colour in the solution should be changed to green, but other than that I like it as it is. Figuring out what Melanie knows is the solution key, which the wording says.

Of course, some people would be hoping for the solution to also reveal how Judith was able to figure out the colour of her hat. I'm not sure if the reasoning should be revealed, but if it were, here's an option (wintersolstice's, slightly adjusted):

About crediting, well, I don't really know, just that it would be nice for readers to see the activity, I suppose. So if you were to say something about that, wintersolstice should receive the credit for the discovery, and Bob & I could be mentioned as confirming it.

Btw, when reading "as shown below" in the puzzle I expected to see a diagram, but there isn't one and "one behind the other" might be better.

Hi wintersolstice and Bob;

I thought you'd never ask, Bob!

Btw, I have one of those, and below is a pic I took of it tonight. It's never worked for me, though!

And...I agree with both of you.

Nice puzzle!

Yes, salem_ohio's {8,8,1} method is easily the most elegant one here, with its simpler, more logical approach:

Rotten apple numbers from the following weighings:
Split 16: W1 (8:8) = 0 or 2
Split 8: W2 (4:4) = 0 or 1
Split 4: W3 (2:2) is the decider: if there's 1 rotten apple left, it can't be split, and therefore a balanced weigh = all good apples, and unbalanced = rotten.

The final (2:2) weigh is either of these two: 2 good vs 2 good...for a balanced result; or 2 good vs (1 good + 1 bad)...for an unbalanced result.

Done in 3 weighings.

My 2-weigh solution of the {7,7,3} is very laborious, being just trial and error. It would be a pity if it was the winner as it would only win on minimum weighs, not technique. However, I've looked at it pretty closely and can't fault it yet.

And it looks like my {7,7,3} solution isn't the only 2-weigh one, either, because that strategy also seems to work for {5,5,7}. That would further sour the {7,7,3) result, as that means there are multiple solutions. I haven't tried my T&E strategy on other groups yet.

This was my reasoning for all possible rotten apple combinations and their respective weighings:

We have {A,B,C} = {7,7,3}.
That gives 2 options of rotten apple combinations: {1,1,3} and {2,2,1}.

Transferring 2 apples from A to C yields the following rotten apple options:
For {1,1,3}, it's {1,1,3} and {0,1,4}.
For {2,2,1}, it's {2,2,1}, {1,2,2} and {0,2,3}.

Weighing A:C is as follows:
For {1,1,3}, it's {1:3} and {0:4).
For {2,2,1}, it's {2:1}, {1:2} and {0:3}.

None of these weighings give a balanced result.

This image may help:

After first weighing A:B in a 6:6:5 combination and then transferring 1 apple from A to C, the second weighing, this time with B:C in a 5:6:6 combination, identifies that the transferred apple is bad.

I tried to use that information to solve the puzzle at the third weighing, but was unsuccessful. It might be possible, but I gave up looking when I saw thickhead's solution.

EDIT: Hi, thickhead. I just read your last post, where you gave the solution to the place where I'd got stuck with the 6:6:5, with the outcome that this combination can be solved in 3 weighings. Well done! I hadn't seen that the opportunity was there to increase a group's bad apple count to 3, which would always result in an imbalance.

Xlnt, thickhead! Can't see anything wrong with that!

I was still at the stage of moving just 1 apple over, and then saw your post.

I'd got 4 min in nothing flat and it all seemed so open, so I was pretty sure there'd be a lower min...but not right down to 2!

Hi, sirluke4, and welcome to the forum!

I have an answer, and it is one of the six possibilities you listed.

Well...looks like that index rebuild has fixed all my OneDrive links! None are broken now.

So I've only had to replace the Kiwi6 links, which I know for sure are broken.

Yes, some are old - from 2015 and before - but some are from 2016, with one as recent as October last year. Furthermore, a post from a couple of weeks earlier - in September - was successful on the same image that refused to change in the October post!!

Riddle me that one, Batman!

Anyway, it's 'good' to hear that I'm not the only one this is happening to. I'll stop pulling my hair out, then.