Multiply any four consecutive positive integers and add 1 to the product. What kind of a number do you get? Will this always happen? If you think so, prove it.

## Identifying Patterns

## Sum of Odd #s is the Squares 2

What is the sum of the first *n* odd numbers? (The first five odd numbers are 1, 3, 5, 7, 9.)

## Sum of Odd #s is the Squares 1

- What does picture A have to do with picture B?
- Continue both A and B for two more steps.
- Find the totals in A. What kind of numbers are they?
- Does B tell you why these numbers are what they are? Explain.
- What is 1 + 3 + 5 + 7 + . . . + 99?

## Formula: Recursive to Closed

If *f*(1) = 2 and *f*(*n*) = *f*(*n* – 1) + *n*, then what’s a formula for *f*(*n*)?

## Calendar Square of Dates: Formula

Take a calendar, and draw a square around any nine dates (it will be a 3 x 3 square, right?) Let *s* be the smallest number in the square. If you add up all the numbers in the square, the answer will always be a multiple of 9. In particular, the sum will be exactly 9 · (*s* + 8). Why?

## Grapefruit Stack (14)

On a special day at the West Side Market, the grapefruit are arranged in a compact stack of filled-in equilateral triangles with 14 grapefruit on each edge of the bottom triangle, 13 on each edge of the one above, and so on all the way up to the top where there is 1 graperfruit sitting all alone. How many grapefruit are in the entire stack?

## Four 4’s: 1 – 20

You are to find expressions for each whole number from 1 through 20, using in each expression exactly four 4�s, plus any mathematical symbols you like: parentheses, plus signs, etc.

Sample: | 1 = | 4 + 4 | . |

4 + 4 | |||

## Four 4’s: 21 – 40

In a previous problem, you found mathematical expressions equalling each of the numbers 1 – 20 using exactly four 4’s and any mathematical symbols you wanted to. Now make 21 – 40 following the same rules.

## Multiplication Pattern 1

Find the pattern for these equations, and then make up one more of your own, using the same pattern.

13 � 62 = 31 � 26

## Squares & Cubes Pattern

Examine the numerical statements in the box. Something is going on.

What would the statement be for 100^{3}?

What would the statement be for *n*^{3}?

Can you prove that it always works?