Creative Counting

One of my favorite challenge activities with students is to ask them if they are older than one million days (hours, minutes, seconds). If so, they work out their age in each of the possible units. They are highly motived with these seemingly simple questions and begin immediately (using a calculator) to find the answers. You might want to find out if you are at least one million days (hours, minutes, seconds) old.

Sometime in 2020, I pondered over a similar question that appeared as a Mathematics Monday problem published by MoMath, the mathematics museum in New York. The question was something like:

To explore these questions, I started by multiplying the various time units as shown above. The answer was easy to calculate so question 1 was answered. Now, what did I notice about the number? It didn’t look particularly interesting, so I began to ‘play around’ with the numbers. My first thought was to break up the numbers in the product so I write each as a product of factors. In particular I broke this down as far as possible and used the prime factors which is a common way to analyze numbers when multiplication is involved. The number of days is 7, a prime number so, it remained unchanged.

One of the first observations is that every prime number less than 10 is listed. I crossed off one of each of these prime numbers. The remaining prime factors are shown below.

Now what? After a little more ponder, I did see that several other non-prime numbers less than ten could be formed. My analysis identified 4 and 6 as shown in the next illustration.

From this, I had identified the numbers, 2, 3, 4, 5, 6, and 7 that could be copied or formed from the prime numbers. What remained? Using three 2s, gave 2 x 2 x 2 = 8. Then 9 came from the 3 x 3, and 10 could be created from 2 x 5.

I had every number 2 to 10! Finely, multiplying by 1 doesn’t change the value, so it could be included as well. That meant that 3,628,800 could be written as the product of the first ten counting numbers:

In mathematics, the product of counting numbers from 1 to any number, say 10 can shortened to 10!. This expression is read as 10 factorial. Note, the exclamation mark (!) is, indeed, part of the notation. And yes, it was chosen because the size of the number gets large surprisingly quickly. To check how quickly the values of the factorials increase, find the values of these expressions:

Factorial expressions appear frequently in mathematics where ‘things’ need to be counted. Every number in the famous Pascal’s triangle is calculated using a combination of factorials. The numbers in this famous triangle come from factorials that are written as fractions (see below). These might seem challenging to calculate, but are usually rather easy. Try these examples. Write out each expression and make sure to cancel as much as you can before multiplying.

Remember, it might be easy to reach for a calculator to work seemingly complex multiplication (or addition, subtraction, or division examples), but many times, the process is easy to do mentally if we just – think!