Puzzles from the Math Midway
Beyond The Century

The Math Midway exhibit "Number Line Tightrope" presents fourteen interesting families of numbers with colorful tags so that you can see the patterns. The physical number line in the exhibit ends at 100, but each of these families goes on much farther -- in many cases, the families go on forever. Can you figure out the next number beyond 100 for each of the families? The patterns can help!

Primes These are the numbers that have no smaller factors other than 1. The last few primes up to 100 are 79, 83, 89, 97. Unlike many other families, there's no simple pattern to the spacing between consecutive primes. In order to figure out whether a number is prime, you have to test it -- for example by dividing by other primes up to its square root. Can you find the next prime number larger than 100? You'll need to make sure the number is not divisible by 2, 3, 5, or 7.

Squares These are the numbers you can get by multiplying an integer by itself. The squares up to 100 are 1,4,9,16,25,36,49,64,81,100. Look at the differences between consecutive squares: 4 - 1 = 3, 9 - 4 = 5, 16 - 9 = 7, and so on, so the sequence of differences is 3, 5, 7, 9, ... Do you see a pattern? Can you use that pattern to find the next square number?

Fibonacci Numbers This is the sequence of numbers you get by starting with 1 and 1 and then adding the last two numbers of the sequence to get the next one, so it goes 1, 1, 1+1 = 2, 1+2 = 3, 2+3=5, etc., or in other words, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89. To find the next one, which is more than 100, you just keep adding.

Triangular Numbers These are the numbers you get by adding up 1, then 1+2, then 1+2+3, then 1+2+3+4, etc. The difference between each triangular number and the one before is always one more than the previous difference between two triangular numbers. The triangular numbers up to 100 are 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91 -- so what's next?

Perfect Numbers These are the numbers which equal the sum of all of their smaller factors. They are few and far between -- in fact, nobody knows how many there are. Only 47 perfect numbers are currently known. The only two less than 100 are 6 and 28, and the next smallest one is...

Factorials These are the numbers you get by multiplying 1, then 1 times 2, then 1 times 2 times 3, then 1x2x3x4, and so on. The ratio of one factorial to the previous one is always one more than the previous ratio. The factorials less than 100 are 1, 2, 6, and 24, so what's the next one?

Powers of Two These are the numbers you get by starting from 1 and doubling over and over again. The powers of two less than 100 are 1, 2, 4, 8, 16, 32, and 64, so what's the first one bigger than 100?

Cubes These are a lot like squares, except you multiply a number by itself three times. The cubes of the first four counting numbers are 1, 8, 27, 64, and those are the only ones smaller than 100. So the next one would be the cube of 5, which is what number?

Highly Composite Numbers These are the opposite of primes, numbers that have more unique factors than any smaller number. A bit like the primes, their spacing does not follow a simple pattern. You have to look for a number which is divisible by lots of smaller numbers. The largest highly composite number less than 100 is 60, which has 12 different factors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. So you need to find the next number with more than 12 factors. Hint: it has 16 factors.

Pizza Numbers These numbers tell you the greatest number of pieces you can cut a pizza into with a certain number of cuts. For example, with one cut you can obviously make at most two pieces, so 2 is a pizza number. The next one is 4, since you can cut at most four pieces with two cuts. Then comes 7, which is the most pieces you can make with three cuts. And so on... the pizza numbers less than 100 are 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92. What do you think comes next? Hint: Compare the pizza numbers to the triangular numbers above. Notice anything?

Cake Numbers These are just like the pizza numbers, except now you are cutting a cake and so you can slice in three dimensions, not just in two. So the cake numbers are bigger: 1, 2, 4, 8, 15, 26, 42, 64, 93... If you want to see an interesting pattern, write down the differences between consecutive entries in this sequence, and then repeat the process on the sequence you got that way (take differences again). Does the pattern you find help you find the next cake number? Have you figured it out?

Pentagonal numbers These are the numbers of dots in larger and larger pentagonal diagrams, just the way that triangular numbers come from dots arranged in a triangle and squares come from dots arranged in a square. The pentagonal numbers less than 100 are 1, 5, 12, 22, 35, 51, 70, 92. You can use the trick of looking at the differences between these numbers to find the next one. What is it?

Constructible Polygon Numbers Some regular polygons, like a triangle, square, or pentagon can be inscribed in a circle. Others, like a regular heptagon (7 sides) or nonagon (9 sides) cannot. This number family includes the numbers of sides of polygons that can be inscribed in a circle. The exact rule for this family is tricky to write down, but the biggest few less than 100 are 68, 80, 85, and 96, and the next one is...

Tetrahedral Numbers These are the numbers you get by adding up triangular numbers, like you were making a pyramid of marbles. Since the triangular numbers are 1, 3, 6, 10, 15, 21, 28, 36, the tetrahedral numbers are 1, 1+3, 1+3+6, 1+3+6+10, 1+3+6+10+15, and so on, or 1, 4, 10, 20, 35, 56, 84, ... what's the next one?

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