Maths: Patterns in numbers

This entry is here for a reason, and that is that some of the patterns mentioned here will pop up later on. Some of the others won’t, but they are still fun.

* The 11-times table: Multiples of 11 up to 9 x 11 are easy, because you just write the first number twice, and 10 x 11 is easy, but can you see a pattern in 121, 132, 143, 154, 165…? There is a pattern there, when you add the outside digits, but how high does it go?

Building up a 9 times table.

* The 9-times table: The table above has five columns which show in five steps how to create the nine times table, knowing nothing more than the order of the digits.
Look at the columns and spot the pattern. In the fourth column, look at the bottom three lines.

* The endings of square numbers: When you multiply a number by itself, that is a square. The first few squares are 1, 4, 9, 16, 25, 26, 49, 64… Are there some digits that are never found at the end of a square number? Why?

* The endings of cubic numbers. Cubic numbers (or cubes (a³) are all created by multiplying a number by itself to get a square, and then again to get a cube. 1³ is 1, 2³ is 8, 3³ is 27 and so on. There is a pattern in the last digit of the cubic numbers. Use a calculator and pen and paper if you need to, but find the pattern.

* Powers of 5: These have a very simple pattern in their last digit. By the time you get to 5⁴ or 625, you should have seen it.

* Powers of 3: The first few are 3, 9, 27, 81, 243… Continue the series and find the pattern.

* The number of digits in powers of 2: As I mentioned earier, as a small child, I used to put myself to sleep by calculating this series in my head: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096… I recall noticing that there were three with one digit, three with two digits, three with three digits. I thought I had something, but I got to 8192, and knew I was wrong, but is there a pattern? I have no idea.

Looking at a 91 times table.

* Now here’s another curious times-table, involving 91, a number that you should remember, because it will turn up again.

Read down the three columns in the products, and you will see an odd pattern developing. Can you find the other interesting things about 91?

The number 91 again

By an odd chance, 91 is the number of naturally occurring elements which exist on our planet. The last of these elements is uranium (element 92), but technetium, element 61, is never found in nature.

A case could be made for saying that there are just 88 naturally occurring elements, given that another three are incredibly rare. Three more elements are so rare that they might as well not be present. Promethium is formed in small amounts in the fission of uranium and two others decay so rapidly as to be vanishingly rare, with less than 600 grams of each in the entire earth’s crust.

At any given time, there will only be about 30 grams of astatine in the earth’s crust, formed by alpha decay of the other rare element, francium. This forms from the radioactive decay of actinium, and its most stable isotope has a half-life of only 22 minutes, so francium is effectively non-existent outside nuclear research facilities.

Incidentally, 91 = 1² + 2² + 3² + 4² + 5²+ 6². If you were so minded, you could probably build a good swindle on this number, but watch out: the mathematicians are watching!

To search this blog, use this link and then use the search box

Another way: use the index!