## Wednesday, May 26, 2010

### OK Everybody, Line Up

In a rectangular array of people, who will be taller: the tallest of the shortest people in each column, or the shortest of the tallest people in each row?

1. 1 | 5 | 9 | 13
2 | 6 | 10 | 14
3 | 7 | 11 | 15
4 | 8 | 12 | 16

I pictured something like this, where 1 is the shortest person and 16 is the tallest person. The tallest of the shortest people in each column is 4, and the shortest of the tallest people in each row is also 4. So they are the same person therefore the same height.

2. The tallest of the shortest!

3. I was going to agree with Dan, but then I made a larger, more random chart and came up with a solution that showed that the shortest-of-the-tallest is taller:

01 | 16 | 08 | 04 | 13
17 | 11 | 02 | 19 | 21
07 | 03 | 18 | 05 | 10
15 | 25 | 06 | 14 | 23
22 | 12 | 20 | 24 | 09

But I have no idea if it would work out that way every time...I'm sure there's some mathematical principal at work here, or something.

4. Oh, and in my chart the tallest of the shortest in each column is 9, and the shortest of the tallest in each row is 16.

5. The s.o.t.t. is the tallest person in his/her row, by assumption. Of the people in his/her row, at least one (call him/her x) (and not to include the s.o.t.t. him/herself) will be in the same column as the t.o.t.s. We know that x either is the t.o.t.s, or, since in the same column, is taller than the t.o.t.s. In turn, the s.o.t.t. either is x or is taller than x (since x is in his/her row.

So we have:

s.o.t.t. >= x >= t.o.t.s

and hence:

s.o.t.t. >= t.o.t.s

Equality is a posibility; A trivial example (numbering by height) is:

1 2

3 4

Or, even more simply:

1

Inequality is also a posibility. Consider:

9 2

1 8
(s.o.t.t. is 8; t.o.t.s is 2)

But s.o.t.t. either is t.o.t.s or is taller there-than.

6. sorry--read "disclude" for "include" in the first paragraph

7. And here I was thinking that the tallest of the shortest people and the shortest of the tallest people was the same...

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