It’s that time of year again and we can play the 2022 NCTM Year Game in our January lessons. Full rules are here; that’s a lot of 2s, multi-digit numbers such as 20, 220, or 0.22 are accepted for 2022 though note that students are encouraged to find solutions using only the single-digit numbers 2, 0, 2, and 2 rather than double digits like 0.2 or 22.. , students are also encouraged to challenge themselves to use the digits in the order 2, 0, 2, 2.
Playing this with younger students has been an opportunity to introduce the factorial function, and we tend to stray into double factorials as students are curious. A good exercise in algebra for your older students – can they find a relationship between the single and double factorial functions?
Excel has a function for computing double factorials, illustrated here. I like to show my students a few examples and see if they can work out what is going on.
Have a look at this article from Wolfram Math World and check this journal article – Henry Gould, and Jocelyn Quaintance. “Double Fun with Double Factorials.” Mathematics Magazine 85, no. 3 (2012): 177–92. https://doi.org/10.4169/math.mag.85.3.177.
And so to number properties of 2022…
We can always check Tanya Khovanova’s Number Gossip site for properties of 2022, the common properties of 2022 are shown here. All Number Gossip properties are detailed here. I have come across these before, but we have not had an untouchable number since 2010; untouchable numbers are those that are not the sum of the proper divisors of any number. A chart of the first 600 untouchable numbers is available from EasyCalculation.com. Here’s a great post from Iva Sallay on Find the Factors – 516 is an Untouchable Number.
2022 is also an iban number – this has amused me for a long time – get your students thinking outside the box with the iban sequence: 1, 2, 3, 4, 7, 10, 11, 12, 14, 17, 20, 21, 22, 23, 24, 27, 40, 41, 42, 43, 44, 47, 70, 71, 72, 73, 74, 77, 100, 101…
How many ways can you write 2022 as a sum of squares? And can 2022 be expressed as a difference of two squares? I came across this idea from a new year (2009) maths item of the month from MEI, we could be more general – are there any years which cannot be written as the difference of two squares? Here’s a superb resource for this problem on Nrich, What’s Possible? Possible questions and approached in class are included in the article with a printable worksheet for students.
We can also look at WolframAlpha for further information on the number properties of 2022 including what 2022 looks like in historical numeral forms. We could use the various historical numerals examples to learn how Babylonian, for example, numerals work. I have successfully used this as an interesting starter for January lessons.
The Babylonian system was a positional base 60 system, though interestingly uses ‘units’ and ‘tens’ symbols to create the 59 symbols needed.
For more on the Babylonian system including how fractions were represented see History of Fractions from Nrich.
We could look back and use the excellent MacTutor History of Mathematics Archive from the University of St Andrews, Scotland. We could check today or any day for Mathematicians who were born or died on that day.
The site is searchable in several ways, including the comprehensive index of History Topics.
Wishing educators and students everywhere a very Happy New Year.