Lab 7: Fun small exercises

Released: 11:59 PM Sunday, July 12th, 2020.

Due:

In this lab, we'll take a break from projects, and solve some fun problems

Part A: Summing digits.

Last week I was looking at a problem that reminded me of something that I learned a while ago:

Old-guy memories

When I was a kid (feels like 25 years have passed… erm… I guess they did? O.o) I learned of a cool trick to quickly check the result of additions: Casting out the 9’s.

This method works like this. When I add 123 and 256 and I get the result equal to 379. I can do the following to validate the result:

Add the digits of the addends, if it goes over 9 go back to zero:
1 + 2 = 3
3 + 3 = 6
6 + 2 = 8
8 + 5 = 13 (-9) = 4
4 + 6 = 10 (-9) = 1

Add the digits of the result, if it goes over 9 go back to zero:
3 + 7 = 10 (-9) = 1
1 + 9 = 10 (-9) = 1

If the two values are different, then the operation is wrong. If both are correct, then we can’t tell for sure :)

Then why is this useful? Because it eliminates off-by-one errors (actually off-by-up-to-8 errors) because the result only repeats if you make mistakes that add up to 9 :)

What reminded me of this?

There is this operation that is known as the digital root. That serves that same purpose.

The digital root is calculated by adding recursively the digits of a number. E.g.:

digital root of 25453 = 1 because: 2+5+4+5+3 = 19 => 1+9=10 => 1+0=1
digital root of 25589 = 1 because: 2+5+5+8+9 = 29 => 2+9=11 => 1+1=2
digital root of 15 = 6 because: 1+5=6
digital root of 123 = 6 because: 1+2+3=6
digital root of 256 = 4 because: 2+5+6=13 => 1+3=4
digital root of 379 = 1 because: 3+7+9=19 => 1+9=10 => 1+0=1

And this can be used to check Maths:

256 + 123 = 379
digitalRoot(256) = 4
digitalRoot(123) = 6
Extra step: 4+6 = 10 => digitalRoot(10) = 1
digitalRoot(379) = 1

Also works with subtraction: 256-123=133

digitalRoot(256) = 4
digitalRoot(123) = 6
Extra step: 4-6 = -2 => No problem just 9-2 = 7, cause of reasons :)
digitalRoot(133) = 7

Multiplication 256*123=31488:

digitalRoot(256) = 4
digitalRoot(123) = 6
Extra step: 4*6 = 24 => digitalRoot(24)=6
digitalRoot(31488) = 3+1+4+8+8=24 => 2+4=6

Division: \(\frac{31488}{123} = 256\) … so … \(31488 = 256\times123\) :D

So what is the deal?

Helpful Tidbits

1. Modulo is great

   Did you ever notice that?

    12345/10 = 1234
    12345%10 = 5
   

2. There is a class named Integer that can help converting String into int You can convert a number to a string, right?

    ""+12345 = "12345"
   

   This converts a string into an int:

    Integer.parseInt("12345") = 12345
   

Part B: Fun with sequences.

Can you guess the next line in the following sequence? :D

1
11
21
1211
?

What about the next in the sequence? :)

What is this? It’s called the Look-and-say sequence. Because if you read it, you can easily tell the next element :)

The first entry is: 1

If you read it out-loud, it’s 1 time the number 1 or 1 1.

If you read it out-loud, it’s 2 times the number 1 or 2 1.

If you read it out-loud, it’s 1 time the number 2 and 1 time the number 1 or 1 2 1 1.

If you read it out-loud, it’s 1 time the number 1, 1 time the number 2, and 2 times the number 1 or 1 1 1 2 2 1.

Ok ONE HINT!!: “123”.charAt(1)