The number 73 can be written as the sum of 73 consecutive integers. What is the product of these 73 integers? (Copyright mathleague.com.)
If you were reading a book about juggling for beginners, and it recommended starting out by trying to juggle 7 balls, you would rightly think that advice was crazy. Even someone who had no idea how to juggle could tell you that starting out with 3 (or even 2!) balls lets you get a better handle on what is going on. Before one tackles complicated problems, it makes sense to try easier, related ones first.
In mathematics, the way we cut a problem that is intimidatingly large or confusingly abstract down to size is to Change or Simplify the Problem. In the problem above, having to deal with 73 integers makes the problem initially seem extremely hard. How would one even approach finding the 73 consecutive integers that add to 73, and how on earth if you found them could you find their product in a reasonable amount of time?
Trying a Smaller Number is a very popular and useful way to change or simplify the problem. As with the juggling, it frequently allows you to get a much better understanding of what is going on. So in the problem above, why not see if you can simplify it by replacing all the 73’s with 5’s. Then, hopefully you can use the insight you have gained with the easier problem to solve the original problem as well. Give it a try!
Another variation on this theme occurs when a problem is stated in terms of variables like $X$ or $Z$, such as:
At Bob's Breakfast Bungalow, the toast buffet involves $X$ kinds of bread and $Z$ kinds of jam. How many different bread-jam pairings are there, assuming that the customers use only one kind of jam each time they make toast?
When stated so abstractly (“$X$ kinds of bread and $Z$ kind of jam”), the problem can be hard to think about. Changing Variables to Numbers is another way to change or simplify the problem. Why not try specific numbers for $X$ and $Z$ to get a better sense of what is going on? It might make sense to try a few different pairs of numbers so that you can be sure that you have the right idea. Once you do, then, as with trying a smaller number, you can use what you have learned to go back and solve the original, more abstract (and likely more complicated) problem.
You add up $X$ randomly chosen positive numbers. How many times larger is this sum than the average (mean) of those numbers? (Your answer may include an $X$ in it.)
Some students stood evenly spaced in a circular formation. They counted off, starting at 1 and continuing by consecutive integers once around the circle, clockwise. How many students were there in the circle if the student furthest from student 19 was student 83? (Copyright mathleague.com.)
Giuseppe likes to count on the fingers of his left hand, but in a peculiar way. He starts by calling the thumb 1, the first finger 2, the middle finger 3, the ring finger 4, and the pinkie 5, and then he reverses direction, so the ring finger is 6, the middle finger is 7, the first finger is 8, the thumb is 9, and then he reverses again so that the first finger is 10, the middle finger is 11, and so on.
One day his parents surprise him by saying that if he can tell them some time that day what finger the number 1,234,567 would be, he can have a new sports car. Giuseppe can only count so fast, so what should he do?
The following two problems are related.
If Wenceslaus wrote a list of $Z$ consecutive odd integers, by how much would the greatest number on his list exceed the smallest? (Your final answer may have a $Z$ in it.)
The sum of $X$ consecutive odd integers is $A$. The sum of the next $X$ consecutive odd integers is $B$. What does $B - A$ equal? (Your final answer will include an $X$ in it.)
How many squares of any size does an 8 by 8 checkerboard have?
A blue train leaves the station at Happyville going East $X$ miles per hour. A red train leaves the station at Nervoustown, $2(X + Y)$ miles away, at $Y$ miles per hour and headed West towards Happyville. (Answers may include $X$ and $Y$.)
How long will it take the trains to crash?
How far away will the red train be from Happyville when they crash? (Your answer may include an $X$ and/or a $Y$ in it)
How far apart will the trains be one hour before they crash?
Yolanda tells Miguel that she can guess any integer he thinks of from 1 to 10 million in 25 yes-or-no questions or less. Miguel says that is ridiculous!
Who is right, and why?
Let ${10^Z} - 1$ be written fully out as a number (and thus with no exponents). Find the sum of the digits of this number. (You will have a $Z$ in your answer.)
What’s the minimum number of non-overlapping triangles into which you can divide a 2008-sided polygon? (Each triangle side must be a segment connecting two of the vertices of the polygon.)
Find two consecutive positive integers where the difference of their squares equals 3747.
Josephine writes out the numbers 1, 2, 3, and 4 in a circle. Starting at 1, she crosses out every second integer until just one number remains: 2 goes first, then 4, leaving 1 and 3. As she continues around the circle, 3 goes next, leaving 1 as the last number left.
Suppose Josephine writes out the numbers $1, 2, 3, 4,…, n$ in a circle.
For what values of $n$ will the number 1 be the last number left?
In the ordered sequence of positive integers: $1,2,2,3,3,3,4,4,4,4,...,$ each positive integer $n$ occurs in a block of $n$ terms.
How many terms of this sequence
are needed so that the sum of the
reciprocals of the terms equals 1000?
(Copyright mathleague.com.)