Posts Tagged ‘math’
You now have permission to be a numbers snob. You’re welcome.
The word “ringing” for house phones was replaced by “noise” a long time ago, because it’s just horrible.
The largest known perfect number is (22976220)(22976221-1). Yes, you read that correctly. It’s well over a million digits long. Somehow, I’m betting they didn’t work that one out by hand…
A perfect place is fun to imagine. Would there be a moonwalk full of jello and fireworks for precisely as long as it was amusing or would everyone just be Mary Poppins clones?
Maybe the song was about a Gömböc. A Gömböc that likes to party.
The idea of a Gömböc is that there is one area that is stable and one area that isn’t. Thus, it invariably stabilizes itself.
We can deal with Chinny’s inner ear, muscular, or whatever problem he has that’s causing his falling later! My money is on his love of gravel.
In the modern era, encryption and decryption really is more of a computer thing. Back in the day, though, people had to figure out the patterns and algorithms by hand. Finding common words via such things as double letters were super helpful. However, encryptions were sneaky and would mess with spacing, having changing letters (a ‘w’ may be used to represent an ‘a’ in one location but may be a ‘h’ in the next), and etc. One thing that was cool about decryption, other than the math puzzle aspect of it, was its importance in such things as wars. If someone figured out an intercepted encrypted message, they had a huge advantage.
Ok, so I pretty much just jumped on the Sudoku train. It’s been doing its thing for a while and yet it took me this long to succumb to its numbery goodness.
The rules are simple. There are three 3×3 grids, making it so that there are 9 boxes total and every row and column 9 squares long (I’m going to call the 3×3′s boxes and each individual square a square). It looks like this:
Every row, column, and box each has all the numbers 1-9, meaning that there can’t be more than one of the same number in any of those or it wouldn’t work. That’s the game. If I were on the committee for making Sudoku sound cool I’d be fired, but it is fun. Anyway, I’ll show you how to be awesome at it by solving one step by step. Also, before we start, I’d just like to say that following the ideas I use should make it so that you never have to guess. I’ve been told a few times that sometimes you’ll just have no choice but to guess, but so far I don’t really believe them.
Here’s a puzzle generated by my phone on the hardest setting (yes, I said phone):
The first step is smack talk. “You got nutin’ Count Dooku!”
Ok, step two… This one’s kind of a gimme. The 1 in the middle right box leaves only one option for the placement of the 1 in the bottom right box. These two block off 2 columns and, with the 1 in the top middle box, there is only one place for the 1 in the top right box to go. Two numbers down!
There are a couple of easy 3′s to place as well. There are a lot of 3′s starting off on the board (something to look for) and it makes it so that there is only one spot that the bottom left and top right 3′s can go, as shown below.
There are a couple of easy 4′s too.
And a 5…
And now it starts becoming fun! Look at the top right box. You know that there are 4 numbers missing and can quickly deduce that they are 2, 6, 7, and 9. The 6 seems easiest to get since there are 6′s in other boxes that cross with this one, so let’s get that one out of the way. With basic eliminations, we can get it down to 2 squares. However, the top row in the top middle box is full, meaning that, since the 6 in the top left box is on the bottom row, the 6 in the top middle box must be in the middle row. This means that we can eliminate all but the top row on the top right box and find that the 6 has only one spot it can go.
Now we have three numbers left in that top right box, making process of elimination solutions a strong possibility. The remaining numbers are 2, 7, and 9. Conveniently, the bottom right box has both 2 and 9 in the far right column. This means that they must be in the middle column in the middle right box because the left column is full and, thus, they must be in the left column for the top right box. This leaves only the 7 to go in the center square in the top box.
And now our hard work starts to pay off! You may notice that there is only one number missing in the top row now. The only number left is 8, so we can plug that in. Now there are only two numbers left in that top left box. Those are 1 and 2. There’s a 1 in the middle row of the top right box, meaning that the 1 in the top left box is in the bottom right square and, therefore, the 2 has to go in the middle right square. We’ve now completed one of the boxes!
Now that we know that the 2 in the top left box is in the middle row, we know that the 2 in the top right box can’t be, meaning that it’s in the bottom left square. The last number in the box is 9, so we can plug that in as well.
So, we’ve done some damage. Looking around, we can see that there is a 7 with only one place to go in the bottom left box. With that one in place, the one in the bottom right box becomes a gimme as well.
With those in place, you may notice that the bottom right box only has two squares left. Ooolala! Those numbers are 4 and 8, so let’s look at those numbers real quick. We can see that the 8 isn’t going to be super helpful right now because the ones we know are eliminating rows and columns that are already full. However, looking at the 4, we can get three numbers quite easily. The 4 in the bottom left box has only one square it can go in, since the other columns can be eliminated by other 4′s. With that, we can see the placement of the 4 in the bottom right box and, in turn, the 8.
The bottom left square has an easy number to fit in. The left column is full, there’s one space in the middle column, and the right column is open. This means that if any of the numbers missing from the square are in the right column of a left side box, that number belongs in the bottom left box’s bottom middle square.
The missing numbers are 1, 5, 6, and 9. We’ll see that there’s a 1 in the right column and everything will be hunkey dory.
Now the brain gets to wake up from its nap! Taking a look around, we can see that there are several columns with only a few numbers remaining. There aren’t any numbers that we can get by the convenient criss-crosses, so let’s look at what numbers are missing from the columns. On the far right, 4, 6, and 7 are missing. We can see that 7 can’t go in the middle row, but we can see also that neither 4 nor 6 can go in the bottom row, meaning it must be the 7.
Now that we have that, we have an easy 7 in the middle.
Let’s see if we can finish up that column. We can’t eliminate either space immediately and we can’t use the same logic again, but there are a lot of 4′s on the board. By looking around we can see that the 4 in the center box has to be in the middle row. Thus, the middle row is eliminated on the middle right square and the 4 must be in the top right square.
We can plug in the last number in the right side column, which is a 6. From there, we look to see if the new numbers are useful. From a quick glance, the 4 doesn’t appear particularly helpful, but we can see that the 6 allows us to place the 6 in the middle left box.
Now, the top row of the middle boxes is getting close to being filled. There are three missing numbers, which are 1, 2, and 8. Conveniently, we can do the same thing we did before. We see that there is both a 1 and an 8 in the second column from the left, meaning that the square must be a 2. From there, 1 continues to step it up because there’s one in the second to last column, meaning that the square must contain an 8. The last square, then, is a 1.
We can easily place an additional 1 into the bottom left square of the middle left box by eliminating squares.
From there, there is one number left in the far left column. It’s an 8 and we can plug it in.
An interesting tid bit at this point is that we have the 7 in every box except one. Surely we can finish that off by now… And we can with some eliminations.
We also have all the 1′s for every box except one, which we can also fill in at the point with eliminations.
The row we just place the 1 in is starting to fill up, so let’s see if we can get any more numbers out of it. The missing numbers are 4, 6, and 9. We can get the 4 pretty easily.
Since we have another 4, let’s see what we can do with it. For the middle boxes, both the top and bottom have a 4, but the middle doesn’t. The third 4 must be in the left column and, obviously, can’t be in the same box as the other 4′s. This narrows it down to two squares and, fortunately, we have an additional 4 that eliminates one of those squares and gives us our placement.
I was keeping tabs on that particular square, since the 3 in the middle box had two places it could go and that was one of them. At this point, we can eliminate squares and place our 3.
We can also place a 4 pretty easily.
With that 4 joining the fun, we can see that the middle column is getting low on missing numbers. All that remains (not the band) are 2, 5, and 8. In the first box we can already see that it cannot be a 2 or a 5, so it must be an 8.
And we have two numbers left in that row (second from the top). We can see that the middle right square cannot be a 3 and, therefore, must be a 6. The other one is, then, a 3. Omg omg.
“Good news everyone!” We can finish up the 6′s now. With some eliminations, we can place the 6 in the bottom middle box and, with that, the 6 in the bottom left box.
The 6 in the bottom left box was a little rat. It gave up a lot of information with its placement and pretty much just made us win. There are only two numbers left in that box, which are 5 and 9. The middle right square cannot be a 9, so it’s a 5. The other one is the 9. Then, the bottom row only has one number missing. That number is a 5, so we can put that in as well.
Not only that, but the placement of the 6 also left its row with only one remaining number, a 9. We can place that as well.
From there, we have two numbers left in the bottom middle box. Those are 2 and 8. We can see the center square of the box cannot be an 8, so it’s a 2 and the other square is an 8.
Now we have a whole lot of numbers to work with. One of the ones that is still MIA in a couple of boxes is the 2. We can eliminate squares and place the one in the center box and, with that, place the one in the right box.
There is only one number left in the middle right box now, so we can add that in. The number is a 9. Using the 9, we can see that the 9 in the middle left box is in the center square. There is now only one number left in that row, which is a 5.
The last step is a nice little chain reaction. There’s only one number left in the middle left box (and the second column for that matter), which is a 5. With that in place, there’s only one missing number in its row, an 8. We put that in and see that now its column has only one missing number, which is a 9. The last number remaining must, then, be a 5.
And there we have it! Also, I still don’t know that I believe that it is ever necessary to guess. None of the steps were too crazy either, right? No crazier than the guy at work who just won’t shower, anyway. I hope that was enjoyable, and not just because it was a pain in the butt to put all those graphics in. Go forth and Sudoku!
For those of you that are still confused:
Bet + winnings – previous bets = initial bet
The idea is that if you bet something like $100, you can guarantee earning $100 if you can continue to fund doubling your bet until you win.