July 6 - Casework

July 6 - Casework

Roman Hall

Open Ended

1. Each of the 25 cells in a five-by-five grid of squares is filled with a 0, 1 or 2 in such a way that the numbers written in neighboring cells differ from the number in that cell by 1. Two cells are considered neighbors if they share a side. Compute the number of possible arrangements.

Open Ended

2. The doctor gave Amber ten vitamins with instructions to take one or two each day until she runs out. Compute the number of ways for Amber to take the vitamins.

Solution to Problems 1-2

https://www.youtube.com/watch?v=v9frXTd2UMA

Open Ended

3. Using the figure of 15 circles shown, compute the number of sets of three distinct circles {A, B, C} such that circle A encloses circle B and circle B encloses circle C.

Open Ended

4. Compute the number of ways to arrange the digits 1 to 9 in a 3 x 3 grid such that the numbers increase from left to right in each row and from top to bottom in each column.

Solution to Problems 3-4

https://www.youtube.com/watch?v=og47h15kUNU

Open Ended

5. Compute the number of rectangles in the figure.

Open Ended

6. Eight blue and five orange tiles are to be arranged in a line such that the first tile is blue and every tile is adjacent to at least one tile of the same color. If all thirteen tiles are used, compute the number of possible arrangements.

Solution to Problems 5-6

https://www.youtube.com/watch?v=ieKWJgGwM78

Open Ended

7. The figure shows five rows, each with five bricks. Compute the number of ways to choose five bricks, one in each row, such that any two bricks chosen in adjacent rows are adjacent to each other.

Open Ended

8. The number  $40231$  is a five-digit positive integer that uses five consecutive digits in no particular order. Compute the number of such five-digit integers.

Open Ended

9. Compute the number of ordered triples of integers  $\left(x,y,z\right)$  for which  $x^2+y^2+z^2<3$ .

Solution to Problems 7-9

https://www.youtube.com/watch?v=ZCPbpKUbQvQ