**TOWER 1**

Tower 1 is a game that apparently looks somewhat similar to “Tower of Hanoi”, but has its own distinct rules.

The game is about flipping discs on a single tower. Each disc on the tower (there can be N number of discs) has two faces, each of which is colored blue except for the face of the bottom most disc (touching the base) which is painted orange.

The idea is that every time a certain face of the disc touches the bottom it changes its color to orange.

[If you give numerical weightage to each color, then red= 1 and blue= 0. So when we start the game the default weight of the tower is 1]

Fig1: Initial position [a game of N=3 discs]

Rules of the game: You can flip the top most disc on itself. For every disk below you can flip it only with all the discs above it.

Consider that you have stacked pan-cakes on the frying pan and you are flipping them such that each face is heated (touches) the face of the frying-pan at least once, while maintaining the full stack at all times.

In this case one of the possible moves from figure 1 is as follows:

Fig2: Flipping all the 3 discs together

In this case the weight of the tower becomes 2.

The objective of the game is to get all the faces turn into same color.

Fig3: Final objective reached.

In this case the final score of the tower is 6.

*Note: For a N disc game the highest weight possible is Nx2.*

**Variations to the game:**

Consider a scoring based on the weight of each face of the disc as well as the number of moves.

In that case the score in figure1 is [Number of moves - weight = 0 - 1= -1]

In figure 2 the score becomes [2 – 2 = 0]

Next best move will lead to [3 – 2 = 1]

And so on.

What is the best possible score you can achieve and where should you stop to achieve that score?

This version of the game enables you to get the best score even without finishing the game.

Another variation comes when we ignore the faces and give weights to each disc instead.

In that case the stack looks something like this

And the first time any blue (1) disc touches the base it completely turns orange (1). With a flip similar to figure 2, the stack looks like this.

We can also put constraints where the base disc (the N^{th} disc) must be changed after 2 moves etc.

Tower1 by Arka Roy chowdhury is licensed under a Creative Commons Attribution 4.0 International License.

Based on a work at http://arkaroychowdhury1.wordpress.com/2014/06/02/tower1/.

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