Beginners Problem Solving Skills

Teams of up to 35 students explore mathematical concepts while developing flexibility in solving non-routine problems with multiple solution paths. Problem sets provided by Talent Academy will prepare your students to exceed the rigors of your core curriculum by developing higher-order problem-solving skills.

Here is a sample from the Beginner PSK Level 1 Paper Set.

Problem :- There are 20 points inside a triangle. They are connected by non-intersecting segments with each other and with the vertices of the triangle in such a way that the triangle is dissected into triangles. How many smaller triangles do we have?
Solution :- By using the formula: N = 2n + 1 = 2 * 20 + 1 = 41.

Question :- In a certain football league, the only way to score is to kick a field goal for 3 points or score a touchdown for 7 points. Thus the scores 1, 4 and 8 are not possible. How many positive scores are not possible?
Answer :- One checks directly that the following list of scores up to 14 is the complete list of obtainable scores up to that point: 3, 6, 7, 9, 10, 12, 13, 14. hence, 15 = 12 + 3, 16 = 13 + 3, and 17 = 14 + 3 are all obtainable and so are 18 = 15 + 3, 19 = 16 + 3, 20 = 17 + 3, and so on. Therefore, the positive integral scores which are not obtainable are 1, 2, 4, 5, 8,and 11. Thus, the answer is 6.

Asking ;- 2006 unit cubes are fastened together to form a large rectangular prism with integer lengths. This rectangular prism is painted and then separated into the original cubes. If the number of these unit cubes which have three faces painted is 178, find the number of these unit cubes which have two faces painted.
Ans ;- 2006 can be factored into an m by n by r format in three different ways: 2006 = 2×17×57 = 59×34×1 = 118×17×1. If the rectangular prism has the dimensions 2×17×59 , the number of unit cubes with three faces painted is 8. Since the question states that the number of unit cubes that have three faces painted is 178, this case does not work. If the rectangular prism has the dimensions 118×17×1, the number of unit cubes with three faces painted is 2(m – 2) +2(n – 2) = 262 which also does not work. If the rectangular prism has the dimensions 59×34×1 , the number of unit cubes with three faces painted is 2(m – 2) +2(n – 2) = 178 which does satisfy the given condition. The number of these unit cubes which have two faces painted is (m – 2)(n – 2) = 1824.

Pro :- Chicken McNuggets come in packages of size 6, 9 and 20. What is the largest number of McNuggets which cannot be purchased?
Sol ;- We are able to get 40, 42 = 9*4 + 6, 44 = 20 + 6* 4, and 45 = 9*5. WSo m – 7 is nonnegative number. If a number is in the form of 6m, it is attainable. If a number is in the form of 6m + 1, 6m + 1 = 81 + 6m – 80 = 9*9 + 6m – 20*4, which is attainable. If a number is in the form of 6m + 2, 6m + 2 = 20 + 6m – 18 = 20 + 6(m – 3), which is attainable by using one 20 and (m – 3) 6’s. If a number is in the form of 6m + 3, 6m + 3 = 9 + 6m – 6 = 9 + 6(m – 1), which is attainable by using one 9 and (m – 1) 6’s. If a number is in the form of 6m + 4, 6m + 4 = 40 + 6m – 36 = 40 + 6(m – 6), which is attainable by using two 20’s and (m – 6) 6’s. If a number is in the form of 6m + 5, 6m + 5 = 20 + 9 + 6m – 24 = 20 + 9 + 6(m – 4), which is attainable by using one 20, one 9, and (m – 4) 6’s. We see that all numbers are covered by the combinations of 6, 9, and 20. So we are sure that 43 is the greatest number that cannot be attained.

This is a form of learning based on discovery: to solve the problem, you must both think and compute systematically.

• Beginners Problem Solving Skill