1. Date: Jan 2nd Week
    
   Question : A  A set of identical square tiles with side length 1 is placed on a (very large) floor. Every tile after the first
   shares an entire edge with at least one tile that has already been placed.
   • What is the largest possible perimeter for a figure made of 10 tiles?
   • What is the smallest possible perimeter for a figure made of 10 tiles?
   • What is the largest possible perimeter for a figure made of 2011 tiles? • What is the smallest possible perimeter for a figure made of 2011 tiles?
   Prove that your answers are correct. 
   
   
   
   
2. Question: B  Five circles in a row are each labeled with a positive integer. As shown in the diagram, each circle is connected to its adjacent neighbor(s). The integers must be chosen such that the sum of the digits of the neighbor(s) of a given circle is equal to the number labeling that point. In the example, the second number 23 = (1 + 8) + (5 + 9), but the other four numbers do not have the needed value.
   

18 - 23- 59- 22- 46
What is the smallest possible sum of the five numbers? How many possible arrangements of the five numbers have this sum? Justify your answers.