# Math Genius

## HL 1 Revise problems we went over in class:

1)      At Springfield Elementary School, a shelf contains either 4 math books or 9 spelling books. On 20 shelves at Springfield Elementary, there are 117 spelling books and some number of math books. All 20 shelves are full. How many math books are on the shelves?
Solution:
Dividing 117 by 9, we get 13. So 13 shelves have 9 spelling books on them. That means there are 7 shelves left with math books. Since there are 4 math books per shelf for those shelves containing math books, there must be 28 math books.

2)       Ari, Barry, and Carrie agree on a 4-digit number. Ari said the number is a multiple of 3, 5, and 11. Barry said the number is a multiple of 2 and 7. Carrie said that exactly one of the 4 digits is 9, but none of the digits is 6. What is the number they chose?

Solution:
Start by multiplying all of the numbers 2, 3, 5, 7, and 11 to get 2310. This is the smallest number that’s a multiple of all of the numbers. Now, if we multiply this number by any number greater than 4, we have a number that has more than 4 digits. We then try the four possibilities:

1*2310
2*2310
3*2310
4*2310

Only the last one fits the criteria, so the number is 9240.

3)      What is the greatest multiple of 37 that has exactly four digits?

#### Solution:

The largest possible four-digit number is 9999, so this puts a boundary on the number. I started with 3 * 37, which is 111. If you multiply this number by 10, you get 1110. If you multiply this number by 9, you get 9990. If you add one more 37, you go over the boundary number, so this number must be it.

## HL 3 – Game:

Powers of 2. In class, we went over powers. In the game at the link below, the goal is to reach powers of 2. You do this by using the arrow keys. If a power of 2 is on top of the same power of 2, if you press the down or up arrow key, the two powers of 2 will be summed to get a larger power of 2. If a power of 2 is next to the same power of 2, if you press the left or right arrow key, the two numbers will be summed.

https://poweroftwo.nemoidstudio.com/1024

## https://www.ted.com/talks/arthur_benjamin_does_mathemagic

### HL 9  Complete the following problems:

1) Each letter in the ordered list A, B, C, D, E, F, G, H represents a number. The numbers are not necessarily different, which means that two letters can represent the same number. The sum of the values of any three adjacent letters is 20. When B = 6 and D = 9, what is the value of F?

2) The Funny Book has its pages numbered in the following way: 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5…That is, there is one 1, two 2s, three 3s, four 4s, five 5s, and so on. How many actual pages (including front and back) will the funny book have if it contains all possible pages that are numbered “1” through “20”, but not any that are numbered “21”?

### HL 10 Math fun: Palindromes with multiplication:

1 x 1 = 1
11 x 11 = 121
111 x 111 = 12321
1111 x 1111 = 1234321
11111 x 11111 = 123454321

### Tracy has A quarters and B dimes with a total value of \$3.45. Tracy has more quarters than dimes. How many different values of A can Tracy have?

1)      Solve the following:

1. a) Staci looks at the first and fourth pages of a chapter in her book. The sum of their page numbers is 47. On what page does the chapter begin?
2. b) Different letters represent different digits. AB is an even two-digit number. EEE is a 3-digit number. M has a single digit. Find AB, EEE, and M given than EEE = M * AB

2)      You are given the following pattern: 2, 4, 6, 8…

Person A says, “If the pattern continues, the next number must be 10.” Is he right? If not, why not?

3)      Summing the first x numbers is given by the formula (x*(x + 1))/2. For instance, the sum of the first ten numbers is given by (10 * 11)/2, which is 55. What about summing the first x numbers, where each of the numbers is cubed? That is, what about summing 1^3 + 2^3 + 3^3+…x^3? For instance, the sum of the first three numbers with each of them cubed is 1^3 + 2^3 + 3^3 = 1 + 8 + 27 = 36.

Hint: what happens if you raise the formula for summing the first x numbers to a power?