Dhayam

With a very mixed emotion which I don't know how to even explain amidst the situation because of Corona Virus, I am writing this piece of my observation and study.

Paramapadham (பரமபதம்), is a traditional snake and ladder game placed with Dhaya Kattai (தாய கட்டை). During my school summer vacations, in early 2000s where phone has not reached the middle class, his was one of the options available for time pass and fun. This is believed to be played during the night of Vaikuntha Ekadasi, to stay awake the whole night. Based on the karma of the player, the number of times he gets bitten by snake or raise through ladder depends.  One more interesting fact I would like to remind you is Paramapadham is the 108th Divya Desam as described by Vaishnavaites, which represents Vaikundham. 

Our traditional games, practices, habits have so much science, maths and simplicity in it, which is why slowly we have lost to understand the inner meaning and left those practices. Paramapadham is a board game played by rolling two-four-sided dice, where each dice is denoted as 1, 2, 3 and 6 (0). The played can start his game once if his outcome of rolling the dice is ONE (DHAYAM). (Probability = 1/6)

  

   Fig 1: Choli                                                                     Fig 2 : 6-sided Dice

Fig 3: Dhayam (1)

       Fig 4: Number 2 (1st way)                                                 Fig 5: Number 2 (2nd way)

         Fig 6: Number 3 (1st way)                                                 Fig 7 : Number 3 (2nd Way)

              Fig 8: Number 4 (1st way)                                                   Fig 9: Number 4 (2nd way)

Fig 10: Number 5

Fig 11: Number 6

Fig 12: Number 12

In modern computer science and cryptography, Random Number / PRNG - Pseudo Random Number Generator is term very vital and widely used. After getting to know this term and its properties, I have felt, Dhaya Kattai or Dice is the simplest Random Number Generator which was available for everyone.

Ram : 5, 6, 3, 2, 5, 4, 4, 5, 2, 3, 1, 2, 1, 5, 3, 4, 1, 12, 1, 4, 4, 5, 6, 1, 4, 12, 6, 1, 3, 3, 4, 6, 4, 4, 2, 3, 3, 2, 6, 4, 3, 5, 5, 2, 2, 3, 4, 4, 3, 3, 12, 5, 5, 2, 4, 2, 3, 2, 5, 2, 5, 3, 2, 3, 12, 3, 5, 3, 3, 3, 4, 12, 12, 2, 4, 2, 3, 2, 1, 2, 3, 3, 1, 3, 12, 2, 3, 4, 12, 4, 3, 2, 4, 1, 3, 5, 2, 2, 4, 5, 2, 1, 2, 1, 2, 4, 3

On the first look, we might see same number occurring in consecutive rolls. Just watch carefully the outcomes written in more details for Person 2.

Mani : 5, 6, 4 (3+1), 2, 3 (2+1), 2, 6, 4 (3+1), 4 (2+2), 2, 2, 3 (2+1), 5, 3 (2+1), 5, 4 (3+1), 4 (2+2), 4 (3+1), 3, 5, 5, 3 (2+1), 2, 1, 5, 6, 1, 2, 6, 4 (3+1), 5, 12, 3 (2+1), 5, 1, 2, 2, 4 (3+1), 6, 12, 6, 3 (2+1), 5, 3 (2+1), 4 (3+1), 4 (2+2), 3 (2+1), 1, 6, 6, 3, 4 (3+1),  4 (3+1), 3 (2+1), 12, 1, 2 (1+1), 6, 3, 6, 3 (2+1), 3 (2+1), 3, 3 (2+1), 12, 4 (2+2),  4 (2+2), 5, 4 (3+1), 2,  2, 1, 4 (3+1), 1, 1, 2 (1+1), 6, 3, 2, 4 (3+1), 3, 5, 2, 1, 1, 5, 1, 1, 3, 3, 3, 4 (3+1), 6 , 2 (1+1),  3 (2+1), 2 (1+1), 3, 12, 12, 1, 3, 12, 2 (1+1), 5

To understand the random pattern I also rolled a 6-faced single dice, here is the outcome for that.

Ram : 3, 5, 2, 4, 6, 2, 6, 1, 4, 5, 6, 6, 1, 6, 2, 1, 4, 2, 2, 1, 2, 5, 3, 6, 3, 4, 3, 3, 5, 4, 2, 4, 6, 3, 6, 4, 2, 4, 2, 6, 3, 1, 1, 3, 2, 5, 4, 1, 4, 3, 2, 3, 3, 5, 2, 4, 6, 3, 5, 1, 2, 6, 2, 2, 1, 3, 1, 4, 4, 1, 6, 3, 6, 2, 2, 3, 2, 3, 1, 2, 3, 6, 4, 3, 6, 2, 4, 2, 3, 3, 1, 5, 1, 2, 2, 4, 3, 4, 3, 1, 6, 3, 5, 6, 2, 5, 3, 3, 4, 1, 5, 3, 5, 1, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 6, 3, 5, 1

Mani : 1, 2, 6, 4, 4, 2, 3, 2, 3, 3, 1, 2, 2, 4, 3, 2, 3, 3, 6, 3, 5, 4, 4, 1, 2, 4, 2, 4, 5, 6, 2, 4, 2, 2, 2, 1, 1, 1, 3, 4, 5, 2, 3, 6, 3, 5, 2, 2, 1, 6, 5, 4, 4, 4, 3, 4, 4, 2, 1, 2, 5, 6, 3, 1, 2, 4, 5, 3, 3, 5, 5, 2, 3, 3, 4, 4, 4, 5, 6, 2, 1, 1, 2, 3, 3, 2, 4, 6, 3, 2, 4, 6, 4, 6, 5, 3, 4, 3, 3, 1, 6, 1, 2, 4, 1, 3, 6, 2, 2, 4, 6, 4, 2, 2, 3, 4

Sexagesimal number system (Base-60), was always a topic of interest to me, I could correlate certain things between this game and the number system.

Number 60 is the smallest number that is divisible by 1, 2, 3, 4, 5, 6.  The dice or choli contains the values from 1 to 6. Even in Dhaya Kattai, if we look at the single four-sided piece, we have values 1, 2, 3, 6.

I assume a modulo 6 operation when Dhaya Kattai is used.

Dice 1	Dice 2	Modulo
1	6	7 % 6 = 1
2	6	8 % 6 = 2
3	6	9 % 6 = 3
1	1	2 % 6 = 2
2	1	3 % 6 = 3
3	1	4 % 6 = 4
2	2	4 % 6 = 4
3	2	5 % 6 = 5
3	3	6 % 6 = 6 (0)
6	6	12 % 6 = 12 (0)

Dice 1	Dice 2	Modulo
1	0	1 % 6 = 1
2	0	2 % 6 = 2
3	0	3 % 6 = 3
1	1	2 % 6 = 2
2	1	3 % 6 = 3
3	1	4 % 6 = 4
2	2	4 % 6 = 4
3	2	5 % 6 = 5
3	3	6 % 6 = 6 (0)
0	0	0 % 6 = 12 (0)

The below table shows the modular arithmetic operation over addition (+) for 6

+	0	1	2	3	4	5
0	0	1	2	3	4	5
1	1	2	3	4	5	0
2	2	3	4	5	0	1
3	3	4	5	0	1	2
4	4	5	0	1	2	3
5	5	0	1	2	3	4