# Real Mathematics: Game #3

Chocolate Box

A dear friend of mine started a new business recently. I bought a box of chocolate with the intention of visiting his store. Eventually I did visit his store. Although I could not stop myself opening the box before my visit. In the end I murdered almost half of the box. Sorry matey!

Today I checked what is left in the box and I realized something: Why aren’t same types of chocolates in the adjacent compartments?

Being Neighbors

In order to avoid neighborhood between same types of chocolates, we should be careful not to place them as shown in the photo.

Let’s call this rule “sufficient chocolate density”, or “S.C.D.”.

Question 1: Assume that you have four different types of chocolates and seven chocolates from each type. Could you have S.C.D. in such a box like the following photo?

Question 2: What is the least number of different chocolate types you can place inside this box?

Question 3: In case you have a box like shown below, at least how many different types of chocolate do you need to maintain S.C.D.?

Question 4: If you had four different chocolate types and nine from each type, could you be able to construct a box that has S.C.D.? If yes, what would it look like?

M. Serkan Kalaycıoğlu