CUNY Math Challenge Blog

Solutions, Resources, Further Information & Discussion

Round 4: Problem 4

with 5 comments

Grid

A square grid with 10 rows and 10 columns contains the numbers 1, 2,…, 100 in its squares, as shown to the right. The grid is to be divided into 50 rectangular pieces, each of which contains two squares. Each piece is given a value equal to the product of the numbers that are on it. What is the smallest possible sum of the values of the 50 pieces? (You should prove your answer is the minimum.)

Advertisements

Written by Administrator

April 13, 2009 at 03:03

Posted in Round 4

5 Responses

Subscribe to comments with RSS.

  1. very confusing.
    at first, I was thinking 100*1+99*2…
    but after I read the FAQ, seems like it needs to be 2 adjecent squares..
    so I changed my mind to 100*90+99*89…
    then I felt like this problem is impossible to be that easy…
    can we have more hints?

    ken

    April 25, 2009 at 23:25

  2. (a-b)^2 = a^2 + b^2 -2ab
    so minimizing (or maximizing) the sum of ab is like doing so for the sum of (a-b)^2. (since the sum of the square of the individual entries is constant) So, to minimize, make a-b = 1 for each pair.

    jim

    April 27, 2009 at 07:45

  3. That should have been, to maximize. To minimize you want the difference to be as large as possible, so (a-b)=10

    jim

    April 27, 2009 at 11:18

    • That’s right now, you had me a bit confused for a sec. You want to pair vertical entries, so the minimal sum ends up being 166675 I believe.

      Administrator

      April 27, 2009 at 16:35

  4. Nice puzzle =)

    Jax

    August 26, 2010 at 16:28


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: