Instead of a gif, here is a little peak of "math is crazy and awesome and beautiful." The audio is just quiet music, so you can watch it on mute without missing anything.
Have you been keeping track of your data? Show me!
Ok, so, we will talk about the passcode breaking today. I am going to write a summary of my explanation below, in case anyone is absent or needs review. But I'll be honest, don't just read it. Write things down, and look and think on them, because this is not obvious stuff.
Today
- talk about how we thought about the homework
- talk about factorial
- make a spreadsheet that calculates factorial from a list of numbers, without using the built in function (homework)
Tomorrow
- more spreadsheets, less KA for a while
==== Below this is an explanation of the passcode breaking =====
This is not the most mathematically efficient way to calculate it... But it feels like a good balance between using math to save time, and still being understandable.
- How many 4 digit passcodes are there made up of one numeral?
- How many 4 digit passcodes are there made up of two numerals?
- How many 4 digit passcodes are there made up of three numerals?
With one, it is easy to count: 1. With two numerals, let's say 0 and 1, we list possible values out:
0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101, 1110
and count: 14 values.
With three numerals we either list, if we are patient, or try to figure out a pattern. Since I am not patient, I will try to find a pattern.
In the two , I notice that there felt like there were two missing numbers: 0000 and 1111. With those two included, it's every way to have a 4 digit number made of 2 numerals. So then the 14 looks like 16 - 2... the sixteen possible ways, minus the two that don't work because they only have one numeral.
One way to think of this is that there are four slots for a number to go in: _ _ _ _. If there are 2 digits that can fill each slot, we get 2 ways to fill the first slot, two ways to fill the second slot, 2 ways to fill the third slot, and 2 ways to fill the fourth slot. 2 2 2 2. Since that two means "This could be one number OR another number", and the numbers don't depend on each other, these are the kind of possibilities that we multiply, so we mean 2*2*2*2 possible ways to put 2 numerals in 4 slots. Those 16 ways are listed above: 0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111.
So our answer of 14 came from saying "How many ways can we put 2 numerals in four slots? Now take away the ones that don't have EXACTLY 2 numerals." So for 3 numerals, we will say "How many ways can we put 3 numerals in four slots? Now take away the ones that don't have exactly 3 numerals."
Ways of putting 3 numerals in four slots: _ _ _ _ -> 3*3*3*3 = 81
How many don't have exactly 3 numerals? If we use 0,1,2 as our numerals, there are 3 ways to just use one numeral: 0000, 1111, 2222.
How many ways to just use two numerals? Think about the 14 ways to just use two, from above (0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101, 1110). We can make those 14 out of 0s and 1s, 0s and 2s, or 1s and 2s. So that gives 14*3 = 42 ways to just use two numerals.
So 81 - 3 - 42 = 78 - 42 = 36.
0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101, 1110
and count: 14 values.
With three numerals we either list, if we are patient, or try to figure out a pattern. Since I am not patient, I will try to find a pattern.
In the two , I notice that there felt like there were two missing numbers: 0000 and 1111. With those two included, it's every way to have a 4 digit number made of 2 numerals. So then the 14 looks like 16 - 2... the sixteen possible ways, minus the two that don't work because they only have one numeral.
One way to think of this is that there are four slots for a number to go in: _ _ _ _. If there are 2 digits that can fill each slot, we get 2 ways to fill the first slot, two ways to fill the second slot, 2 ways to fill the third slot, and 2 ways to fill the fourth slot. 2 2 2 2. Since that two means "This could be one number OR another number", and the numbers don't depend on each other, these are the kind of possibilities that we multiply, so we mean 2*2*2*2 possible ways to put 2 numerals in 4 slots. Those 16 ways are listed above: 0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111.
So our answer of 14 came from saying "How many ways can we put 2 numerals in four slots? Now take away the ones that don't have EXACTLY 2 numerals." So for 3 numerals, we will say "How many ways can we put 3 numerals in four slots? Now take away the ones that don't have exactly 3 numerals."
Ways of putting 3 numerals in four slots: _ _ _ _ -> 3*3*3*3 = 81
How many don't have exactly 3 numerals? If we use 0,1,2 as our numerals, there are 3 ways to just use one numeral: 0000, 1111, 2222.
How many ways to just use two numerals? Think about the 14 ways to just use two, from above (0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101, 1110). We can make those 14 out of 0s and 1s, 0s and 2s, or 1s and 2s. So that gives 14*3 = 42 ways to just use two numerals.
So 81 - 3 - 42 = 78 - 42 = 36.
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