Hey guys, there is this game called "Tower Stack" or "Tower Bricks" or "Tower Blocks" (among other names) and you can play it on Facebook, or MindJolt Games, or Brothersoft Games (etc).
Here is a screenshot to show what I am talking about:
What you have to do is:
- - build tower as tall as possible
- - blocks swing at top and you click mouse to drop them
- - block will fall after you click and it MUST land on top of the block that you dropped previously (otherwise you lose a "life" - you have 3 "lives")
- - if you drop PERFECTLY on top, you get bonus points
- - as the tower gets bigger, it starts to shake back and forth making it harder to drop the blocks on top
Okay, let's do the first block. Just drop it anywhere on the platform as shown below.
Now drop the second block on top (see image below).
You now have 200 points, 100 from dropping the first block, and 100
from dropping the second block. Notice the 2 on the bottom left of the
screen - that records the # of blocks dropped so far. Theoretically, you could go on forever dropping the blocks in this fashion scoring 100 points per drop. But as mentioned above you could get bonus points if you drop it perfectly on top as shown below:
Here we got 200 points instead of 100! Let's do another perfect drop:
Wowzers! 250 this time, instead 100 or 200! If we mess up it goes back to 100 as shown below:
Get 4 in a row and it's 300 points for that 4th block:
Get 6 in a row and it's 400 points for that 6th block:
So let's do some math! Basically the scoring works as follows:
Imperfect drops score:
100 points
Perfect drops score as follows:
(# perfect blocks in a row, score for that block)
(1, 100)
(2, 200)
(3, 250)
(4, 300)
(5, 350)
(6, 400)
(7, 450)
(8, 500)
...
(n, 50n + 100)
What this means is that if you have n-1 perfect drops in a row, on the n'th perfect drop you will score an amazing 50n+100 points! (Note that for simplicity, we take the convention that the first drop was perfect).
Obviously one can figure out the optimal strategy now. If you keep getting imperfect drops, then you only get 100 points per drop:
But if you keep getting perfect drops you will score HUGE points on each drop. Below I had 19 perfect drops, so on that drop I got an outstanding 1050. If I get another perfect drop after that, the next one will be worth 1100.
It gets harder as you get lots of blocks. In the final image below I made it up to 132 blocks, but I didn't score that high because I just couldn't get in the rhythm of successive perfect drops.
So the question you should ask yourself is:
how high of a score can you get?Well, let's assume you only have 100 blocks to drop until it becomes too hard and the game ends.
(i) If you do 100 imperfect drops, each drop will be worth 100 points. Thus, the score would amount to:
10,000 points.
(ii) On the contrary, what if you have a perfect game so far. Then you would receive the following points:
100 + 200 + 250 + 300 + 350 + 400 + 450 + ...
Can you see what the last number will be in this sum?
If you said 5100 then you are right! This is the number of points you would get on the 100'th drop, and you can use the formula I presented to you above: 50n + 100.
Do you remember how to add up sums of numbers? Let's do it in general. Let's say you get n perfect drops in a row. How many total points would you get from those n drops? The total is:
This just follows from the sum formulas that you may have learned in either high school or university (note that the formula 50n+100 only works for n>=2, that's why we started the summation at 2). It's okay if you forgot the sum formulas or don't follow all the steps. The main point is that,
if you get n PERFECT drops in a row the TOTAL number of points you will score is 25(n^2 + 5n - 2). Let's do an example. If n=5 then the total score you will have after 5 perfect drops is: 1200.
If you get 100 perfect drops in a row (which is possible!) then your total score will be:
262,450
That's an amazing score! I checked facebook and people have like 1,600,000, which seems totally impossible, but if you get 200 perfect drops in a row you're looking at a score of over 1,000,000. Now, mathematically it's possible, but come on!! Who is going to play a game for that long and be that dedicated to get so many perfect drops! I call cheats/hacks! (if it's a real score then I'm truly jealous)!
It was created in 2006 by some guy named Jay Horowitz in Ohio. You can buy it at Barnes and Noble and some other places. In what follows we briefly describe how to solve it...
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