Having done the impossible

Earlier this year, in the beginning of  summer, I challenged my colleagues to do the impossible. Five of us participated and also gave feedback to the following questions:

  1. What did you choose as your impossible?
  2. Did you achieve your goal?
    a) What helped in reaching your goal?
    b) What was difficult?
  3. What was the effect of you efforts to do the impossible on you?
  4. What was the effect of reaching / not reaching your impossible?
  5. What would be you next impossible?

This was no scientific experiment and the results hardy allow any statistically reliable inferences. Anyway, I take the results as anecdotal evidence of the power of striving to do the impossible. In short, the experiences can be summarized in four concise points.

  • Be disciplined, then the impossible becomes possible.
  • Reducing something in your life can also bring you forwards.
  • If something needs to be changed, change it.
  • In the end you will have learned a lot about yourself.

Experiencing and witnessing the importance of these four points personally is in my opinion crucial, if you want to change your life. Therefore, even if this is nothing new, it was extremely valuable to see how small steps and changes can eventually lead to big achievements

 

Below I have summarized in more detail the answers to questions 1 to 5.

What did you choose as your impossible?
Most impossibles were sport activities, either adding or reducing a relatively small amount of sport into the daily schedule. Bringing joy to others and changing eating habits were also chosen as personal impossibles.

Did you achieve your goal? What helped in reaching your goal?What was difficult?
Most participants reached their goal either fully or nearly completely, missing their target on just a few individual days. A good plan, strong resolution, clear scope and small steps helped in reaching the targets. External coaching also helped. Re-starting after a break was difficult, as was introducing completely new habits without any connection to previous ones.

What was the effect of you efforts to do the impossible on you?
The process of going after the personal impossible showed that impossibles are often well within our grasp. Setting and reaching intermediate goals also motivates. Sometimes reducing something in your life can be very beneficial.

What was the effect of reaching / not reaching your impossible?Increased self-awareness, learning to know yourself a bit better, increased calmness and happiness. Reaching something is mainly a matter of consistence, even if the steps taken are small. Changing something that is not working is also an important step.

What would be you next impossible?
Learn to make decisions more quickly, fasting, learn or continue to play an instrument.

 

Returning to mathematics and a proof on limits 1/2

As may be apparent from my latest blog entries, I have spent a decent part of my spare time lately on game theory. In addition to that, I have also been reviewing my university mathematics books, actually the one from our very first course, and at the time of writing this I am fascinated by sequences, series and limits.

During my university studies I did not really understand the beauty and essence of mathematics. Consequently, I did not put in the hours I should have, which I regret. The good news is that it’s never too late to start, so I took out my university math book, already last summer actually, and have occasionally studied the material from the beginning and concentrated on understanding the concepts and constructing proofs. I was pleasantly surprised by two things. First, I could actually understand the provided proofs and construct some of my own, and second, I was surprised how my attitude towards and perception of university level mathematics had changed: I could appreciate the thoroughness and logical flow of proofs and the joy of finding these properties of numbers and sequences. I have also understood better, how “inexact” methods, e.g. estimating a series from above, are permissible, often powerful methods that can provide exact answers.

In this and the following post I provide in total two proofs of my own, based on the practice problems from my old mathematics book. And yes, I must learn to use LaTex if I am to publish more of these posts with mathematical notation.

20160518_Proof1_1 20160815_Proof1_2 20160815_proof1_3

Nim and simple mathematical proofs 3/3

In this post I present to you yet another game, Northcott’s Game, and another example of how new problems can be solved using experience and knowledge gained from previous ones. Again, the post is based of Thomas S. Ferguson’s book Game Theory.

Northcotts’s Game is a game where black and white pieces are placed on a checkerboard, one piece of each color on each row (see Picture 1 below). The players take turns in moving their piece on a single row and only on a single row per turn. A piece cannot jump over another piece. The last player to move his piece wins. In its essence, this game is about blocking the other player’s pieces in such a way, that your last move blocks their final piece, ensuring you the win. But how to do that? It turns out that knowing how to play Nim is a huge advantage.

 

Picture 1: Northcott's Game, start position.
Picture 1: Northcott’s Game, start position.

 

 

 

 

 

 

 

Winning in Northcott’s Game

Before going deeper into Northcott’s Game we’ll recall the definition of a combinatorial game. A combinatorial game is such that (see Ferguson’s Game Theory, Part 1, chapter 1, page 4):

  1. It is played by two players.
  2. There is a usually finite set of possible positions in the game.
  3. The rules of the game specify which moves are legal. If both players have the same options of moving from each position, the game is called impartial. Otherwise the game is called partisan.
  4. The players alternate moving.
  5. The game ends when a position is reached from which no moves are possible for the player whose turn it is to move. Under the Normal Play Rule, the last player to move wins. Under the Misère Rule the last player to move loses.
  6. The game ends in a finite number of moves no matter how it is played. (Ending Condition)

It has to be noted that Northcott’s Game is not a variant of Nim. More specifically, it is a partisan game, since both players have distinct moves. Furthermore, Northcott’s game does not satisfy the Ending Condition, since it is possible to keep playing forever. Thus, Northcott’s Game is not a combinatorial game due to not obeying the Ending Condition, but is still related to them. Therefore, knowing how to play combinatorial games, Nim in particular, is helpful in winning in Northcott’s Game.

As often in game theory, we start from the end, using backward induction, and ask ourselves, how the end of the game would look like. To make this easier, Picture 2 presents an abbreviated version of the game in Picture 1, showing only the top 3 rows.

Picture 2: Abbreviated Northcott's Game, start position.
Picture 2: Abbreviated Northcott’s Game, start position.

 

 

 

 

Picture 3 shows a “near penultimate” final position of the game in Picture 2, where the white player has moved his piece in the lowest row next to the black piece in the same row. It is now black player’s turn, and he can only move right, leaving the white player with the final move. In this final move, the white player “blocks” the black player’s last piece and wins the game.

Picture 3: Abbreviated Northcott's Game, "near penultimate" position.
Picture 3: Abbreviated Northcott’s Game, “near penultimate” position.

 

 

 

 

It is clear that the position presented in Picture 3 is a potential P-position. Our next question is, how to get into this penultimate P-position, and this is where Nim steps in.

Play first Nim, then win in Northcott’s Game

The game before the penultimate P-position in Northcott’s Game can be interpreted as Nim with the following parallels:

  • Each row is a pile of sticks.
  • The number of empty squares between the white and the black piece is the number of sticks in a pile.

Now the target is to make the Nim-sum of the squares between the two pieces on a single row zero to reach a P-position, eventually the terminal position. The terminal position in this game is a position where the pieces are next to each other in each row, since then the number of squares between the pieces will have been reduced to zero on each row. The player to have moved last before this position will win, since the next player can only move towards his end of the row, and then the next player, the one who reached the terminal position, can always cover this distance, eventually making the last move and winning the game. This is generalizable to any number of rows.

Here we also see why Northcott’s Game could go on forever. While in Nim a player must always remove a stick from one of the piles, thus eventually leaving no piles on the table, in Northcott’s Game the players could keep moving their pieces back and forth without even trying to block one another. This kind of play would be pointless, but is possible, and therefore Northcott’s Game does not obey the Ending Condition for combinatorial games.

Like Nimble, Northcott’s Game is a good a example of how to use existing models and knowledge to solve new problems, and how existing solutions can be used to represent a solution to a novel problem as a combination of existing solutions, executed as steps one after another. I hope you will have as much as fun with these games and solving them as I am having.