Hand computation of Laver tables records

How far are you able and willing to compute classical Laver tables by hand?

Top classical Laver table hand computations:

The classical Laver tables can be computed fairly quickly by hand given enough practice. I was able to compute the entire $64\times 64$ classical Laver table in 11 minutes and 44 seconds. That amounts to about 11 seconds for each row in the $64\times 64$-classical Laver table, and a little more than a second for each entry in each row (I only needed to compute the first period in each row since the classical Laver tables are periodic). The records will be displayed in order of their size, then the time it took for the person to compute the tables, and then which tables are most recent.

Bear with me as I wait for people to submit classical Laver table computation records

  1. Joseph Van Name: $64\times 64$-classical Laver table, 704 sec, 1/4/2016.

Top final matrix hand computations:

The final matrices $FM^{-}_{n}$ and $FM^{+}_{n}$ and the classical Laver tables $A_{n}$ encode all the information necessary in order to compute in the generalized Laver table $(A^{\leq 2^{n}})^{+}$ for all natural numbers $n$. The final matrices can be computed by hand, although it will take a lot longer for one to compute the final matrices than it would for one to compute the classical Laver tables. Here are the records for final matrix computations.

Bear with me as I wait for people to submit final matrix computation records

  1. Joseph Van Name: 8×8 final matrix, 4 min and 2 sec, 1/4/2016.

Submitting Laver table computations.

If you have computed a classical Laver table or a final matrix and you would like to be included on this page, then please e-mail me a scanned copy of the classical Laver tables or final matrices that you have computed by hand. Be sure to also include the name which you would like me to post on this page. Please mention the amount of time that it took you to compute the tables if you have recorded the amount of time with a stopwatch.

Tips and tricks for computing classical Laver tables by hand.

There are several techniques that allow you to more quickly compute classical Laver tables. The basic techniques will be listed first and the more advanced techniques will be listed last.

  • When computing your first Laver table, say the $8\times 8$-Laver table, compute the $8$-th row (which is the bottom row) first. Then compute the 7-row and so on until you complete the table. Compute each row from left to right by using the fact that $x*1=x+1$ for $x<8$ and $x*(y+1)=(x*y)*(x*1)$ whenever $y<8$.
  • When computing a row in the classical Laver table, since the rows in a classical Laver table are periodic, one only needs to compute one period in the row in the classical Laver table.
  • Compute the classical Laver tables using hexadecimal instead of using the decimal number system. It is much easier to compute by hand the classical Laver tables using hexadecimal instead of using a decimal number system even if you have not computed in hexadecimal before. When computing classical Laver tables in hexadecimal, you only need to be able to count in hexadecimal, and you do not need to be proficient at adding and multiplying in hexadecimal.
  • If one uses 33 lined notebook paper rather than the 26 lined notebook paper, then one can neatly fit 32 rows of the Laver table on a single sheet of paper which is much easier to look up than when one uses 26 lined notebook paper.
  • Use the homomorphisms between classical Laver tables to aid in computation.