[Guest]

A squad of soldiers, none of whom are of the same height, is mustered in ranks on the parade ground.

Each file must have a shortest soldier. Of all of these shortest soldiers, one must be the tallest (TS.)

Each row must have a tallest soldier. Of all of these tallest soldiers, one must be the shortest (ST.)

Which soldier is taller, the shortest of the tallest soldiers or the tallest of the shortest soldiers, or can't one tell?

(Usually the rear rank of such a formation is incomplete. This is immaterial.)

[Guest]

I would think that the ST would be taller.

Then again, it's all in how they arrange the ranks.

Edited August 2, 2009 by Giga

[bonanova]

Shortest of the tallest is always taller that the tallest of the shortest.

Working on a proof.

[Guest]

I think that TS can equal ST.

See the following matrix

1 22 15 19

2 16 12 18

3 17 20 21

4 11 13 14

TS=ST=14

Edited August 2, 2009 by bonanova
Spoiler added.

[bonanova]

Here's a 3x3 array where they're equal.

6 7 4

9 5 1

3 8 2

Shortest by row: 4 1 2; Tallest is 4

Tallest by column: 9 8 4; Shortest is 4

The number must be both the smallest of its column and the largest of its row.

The larger the array, the less likely this will happen.

[Guest]

Here's a 3x3 array where they're equal.

6 7 4

9 5 1

3 8 2

Shortest by row: 4 1 2; Tallest is 4

Tallest by column: 9 8 4; Shortest is 4

The number must be both the smallest of its column and the largest of its row.

The larger the array, the less likely this will happen.

Oh, yes.

Edited August 2, 2009 by jerbil

[Guest]

ST will be taller than TS. If we was tall enough he would be categorized as tall

[Guest]

Just from observation I think they will always be equal. I am tired though, and thus prone to simple errors.

[Guest]

You will note Bonanova's counter-example. If one adds the condition that ST and TS are not the same soldier, then the enclosed attachment yields a simple solution.

Platoon Answer.doc

[Guest]

Okay I see how it works. Thanks for the clear explanations.

I was thinking about how it was in boot camp. We all lined up tallest to shortest. Then the first four were broken off to make the first rank with the tallest on the left. The next four made the second rank etc. So the TS and the ST would be in the back left corner. In this sorting they would always be the same person.

[Guest]

Given an example squad where an integer is the person's height in "units":

1st row: 20 19 18 17 16

2nd row: 15 14 13 12 11

3rd row: 10 9 8 7 6

The shortest in each file are all in the third row, making 10 the tallest of the shortest (TS)

The tallest in each row are all in the left file, making 10 the shortest of the tallest (ST)

ST = TS

I guess this means you can't tell. In two other examples I did the ST was taller than the TS.