Talk:Crazy J

How did you find these expressions?

I am wondering how you found these expressions. Some of them seem shorter than expected. For example, suppose I want to derive S from J and I. I look at the formula Sxyz=xz(yz). I want to make xz(yz) start with the same thing in xz and yz, so I can use J. One way to do this is Tzx(Tzy)=Tzx(TzTy)=J(Tz)xyT. I know that T=JII. Then, I want to bring Tz out of the parentheses. I could use B, but that would be complicated. An easy way to do this is to use IJ(ITz)xyT=JIJzTxyT. Then, I want to use permuters to bring the xyz to the end. Here are some birds that will help.

R₁xyzw=ywzx

R₂xyzwv=ywvzx

R₃xyzwvu=yuwvzx

It is easy to see that R₁=JR, R₂=JR₁, R₃=JR₂.

JRxyzw=Rx(Rży)w=Rzywx=ywzx

JR₁xyzwv=R₁x(R₁zy)wv=R₁zyvwx=ywvzx

JR₂xyzwvu=R₂x(R₂zy)wvu=R₂zyvuwx=yuwvzx

Therefore, R₁=J(J(JII)), R₂=J(J(J(JII))), R₃=J(J(J(J(JII)))). Now, we can bring z to the end of JIJzTxyT using R₃ to get R₃T(JIJ)yTxz. Then, we can bring y to the end with R₁ to get R₁xT(R₃T(JIJ))yz. Then, we can bring x to the end with R₁ to get R₁TR₁(R₃T(JIJ))xyz. Then, S=R₁TR₁(R₃T(JIJ)).

R₁TR₁(R₃T(JIJ))xyz=R₁xT(R₃T(JIJ))yz=R₃T(JIJ)yTxz=JIJzTxyT=IJ(ITz)xyT=J(Tz)xyT=Tzx(TzTy)=Tzx(Tzy)=xz(yz)

Therefore, S=J(J(JII))(JII)(J(J(JII)))(J(J(J(J(JII))))(JII)(JIJ)). However, this has 26 letters, and your expression has 19 letters. How did you find your expression?

By the way, I have reinstated V* in your list. The correct formula is V*xyzw=xwyz. You originally had xzyw, which is a cardinal extended to 4 variables (Cxyzw=xzyw). Your expression was equivalent to RRR=J(JII)(J(JII))(J(JII)). V* can be derived from R₁ as R₁R₁R₁R₁.

R₁R₁R₁R₁xyzw=R₁xR₁R₁yzw=R₁yR₁xzw=R₁zxyw=xwyz

Therefore, V*=J(J(JII))(J(J(JII)))(J(J(JII)))(J(J(JII))). Is this the shortest expression?

For the shortest ones I use bruteforce (I can publish the source code if you're interested, it's in C++). Your approach for manual IJ abstraction elimination is similar to mine which is described in the article (section 5.4). Regarding V*: I just started my bruteforcer, and it quickly found this (length 15): J(J(J(JII)(J(J(JII))))I)IJI. --Blashyrkh (talk) 13:52, 27 June 2026 (UTC)