# Which is heavier, the heaviest of the light or the lightest of the heavy?

200 eggs are arranged on a dimpled table in twenty columns of ten. All the eggs have been weighed on an analytical balance and stamped with their unique weight.

From each 20 columns, the lightest is chosen and among these "light" eggs the heaviest is colored red and all eggs are put back to their original places.

Now, from each 10 rows, the heaviest is chosen and among these "heavy" eggs the lightest is colored blue and all eggs are put back to their original places.

If both eggs are different, which egg is heavier, blue or red?

This question has been edited drastically from the original which explains the different answers/bickering/downvoting.

[The question was originally asked in an incorrect form, which is why there are two other answers that probably don't now appear to make sense. I'll give the answer to the question as it now appears, and then the answer to the question as it originally was.]

Suppose the heaviest-of-the-light is in row a and column b, and the lightest-of-the-heavy in row c and column d. Then the HOTL is lightest in its column and hence lighter than the egg in row c and column b; the LOTH is heaviest in its row and hence heavier than the egg in row c and column b; hence HOTL is lighter than LOTH. In other words: the blue egg is heavier than the red one.

[The question originally had some number of "lines" -- no distinction between rows and columns -- and took the heaviest and lightest eggs from each line. Here is what happens then:]

The heaviest-of-the-light can be heavier. Let's say the eggs have weights 1,2,...,200 in some units. Let one line have the eggs with weights 191,192,...,200, and arrange the others however you like. The lightest egg in that first line has weight 191 and is heavier than any egg in any other line.

The lightest-of-the-heavy can be heavier. Again, give the eggs weights 1,2,...,200. Now put eggs 181,182,...,200 into different lines, and again arrange the others however you like. The heaviest egg in each line has weight at least 181, which is heavier than any egg that isn't heaviest-in-its-line.

(The question originally had 10 lines of 20 instead of 20 lines of 10. I've adjusted the numbers above for the new version, but the same principle works no matter how many lines, and no matter how many eggs per line, we have.)

• That was a quick up vote for having been posted 31 seconds ago. Speed readers... – user39732 Aug 26 '17 at 1:41
• It looks to me -- though I haven't read either with the care they deserve -- as if both Mike's and Rand's answers are also correct, and I think your objections to them are wrong. But I thought it was worth (1) being clearer and more explicit about the constructions and (2) making the connection to the different puzzle you may possibly have been thinking of. – Gareth McCaughan Aug 26 '17 at 1:44
• I have now read them (and your comments on them) more carefully, and I still think they are right and you are wrong. I am far from infallible, so maybe I've misunderstood something, but if so then three people who are pretty good at these things seem to have misunderstood in similar ways and it might be a good idea for you to clarify exactly what we have done wrong. (More explicitly than e.g. saying "Your logic is two eggs per line", which I still don't understand.) – Gareth McCaughan Aug 26 '17 at 1:52
• I goofed, it should read twenty lines of ten. You're right. – user39732 Aug 26 '17 at 1:52
• It doesn't matter whether it's 10 lines of 20 or 20 lines of 10, so far as I can see. – Gareth McCaughan Aug 26 '17 at 1:53

## Solution

(I'm guessing Gareth has already solved this, since his answer has been accepted, but I haven't looked at his answer before solving it independently.)

Lemma: given any row R and column C, the lightest egg in C is lighter than (or equal to) the heaviest egg in R.

Proof:

row R and column C must intersect at some egg, which is heavier than (or equal to) the lightest egg in C and lighter than (or equal to) the heaviest egg in R. QED.

the blue egg is always at least as heavy as the red egg.

## Previous solution

(This was my original answer, posted before the changes to the question which invalidated it by altering the problem.)

Could be either.

For example:

• if the eggs in the $n$th line weigh $20n+1,20n+2,\dots,20n+20$ for $n=1,2,\dots,10$, then the red and blue eggs weigh

$20\times10+1=201$ and $20\times1+20=40$ respectively, so red is heavier;

• if the eggs in the $n$th line weigh $n,n+10,n+20,\dots,n+190$ for $n=1,2,\dots,10$, then the red and blue eggs weigh

$10$ and $1+190=191$ respectively, so blue is heavier.

• This is also what Rand is doing, so far as I can see. In Rand's first scenario, L = {21,41,...,201} so red = 201, and H = {40,60,...,220} so blue = 40, so red > blue; in his second, L = {1,2,...,10} so red = 10, and H = {191,192,...,200} so blue = 191, so red < blue. – Gareth McCaughan Aug 26 '17 at 1:47
• Yes, I admitted I made a mistake, but I still want the correctly edited puzzle to be here for other puzzlers. Could you delete your answer, so I can repost the correctly edited question, or is this just a bust/loss? – user39732 Aug 26 '17 at 1:59
• Your logic was correct for the initial post. Sorry. – user39732 Aug 26 '17 at 2:23
• @uwnojpjm I've now fixed my answer to solve the new edited puzzle. If you were the person who downvoted, I hope you might consider reversing :-) – Rand al'Thor Aug 26 '17 at 11:12
• @Gareth Thanks for the support :-) Interestingly, now that I've looked at your answer, we've both solved this by slightly different methods (although both based on the same underlying idea of an L-shaped comparison). My lemma looks stronger than the required result you've proved, but actually they're equivalent. – Rand al'Thor Aug 26 '17 at 11:15

It's impossible to tell. Let $a$ and $b$ be numbers, and suppose the egg in line $i$ and position $j$ weighed $ai+bj$, where $i,j$ are indexed from $0$. Then the red egg will weigh $9a$ and the blue egg will weigh $19b$, so it depends on whether $9a$ or $19b$ is larger.

• @uwnojpjm Let's say one row has the 20 lightest eggs, and another has the 20 heaviest. The first row would have the lightest heaviest egg (blue) and the last row would have the heaviest lightest egg (red). In this case, the red egg is heavier than the blue one. (In most other cases e.g. equal distribution the blue one is heavier than the red one). – somebody Aug 26 '17 at 1:08
• Mike's reasoning allows the weights to be non-unique but doesn't require them to be. We can only have ai+bj = ai'+bj' if a(i-i') = b(j'-j); if a and b are coprime then this requires that a divides j'-j and b divides i-i'; if also a is bigger than 10 (the number of lines) and b is bigger than 20 (the size of each line) then this implies i=i' and j=j', so the weights are all different. And we can easily take a,b large and coprime in such a way as to make either of 9a, 19b larger than the other. – Gareth McCaughan Aug 26 '17 at 1:50
• Your logic was correct for the initial post. Sorry. – user39732 Aug 26 '17 at 2:23
• @uwnojpjm Ah, I see what the confusion was. The revised puzzle is pretty interesting! – Mike Earnest Aug 26 '17 at 3:04

If the red egg and the blue egg are in the same row, then the blue egg weighs more because the blue egg is heaviest in that row.

If the red egg and the blue egg are in the same column, then the blue egg weighs more because the red egg is lightest in that column.

If the red egg and the blue egg aren't in the same column or row, then the egg that shares the same column as the red egg and the same row as the blue egg is lighter than the blue egg but heavier than the red egg because the blue egg is the heaviest in that row, and the red egg is the lightest in that column.

Therefore, the blue egg is heavier than the red egg in every case.