Are there any Turing Machines for which the Universal Turing Machine does not work?

Yes, there are.

First of all, if you had already read the page that answered the question How can you have two different Turing Machine Numbers for the same Turing Machine?, then you know that most Turing Machines have infinitely many Turing Machine Numbers. Anytime you have a Turing Machine Number, you can construct a larger equivalent Turing Machine Number for that same Turing Machine by simply inserting another "optional leading zero" into the binary number just before any machine instruction.

Then, if you also already read the page that answered the question Does it matter which Turing Machine Number is used when you run it with the Universal Turing Machine?, then you saw three examples of Turing Machine Numbers: 177,642; 353,770; and 355,274. All three of theses numers are Turing Machine Numbers of Sir Roger Penrose's UN+1 Turing Machine.

The first number (177,642) is the Turing Machine Number you would get if you used the Notation Converter (that you can dowload from the home page of this website) to convert the Machine State instructions into various numerical notations. The Notation Converter always gives you the smallest possible Turing Machine Number for any Turing Machine. This Turing Machine Number works very well with the Universal Turing Machine.

The second number (353,770) is also a Turing Machine Number of the same UN+1 Turing Machine. It was constructed by inserting an "optional leading zero" into the middle of the first Turing Machine Number when written in Binary Notation. This Turing Machine Number works very well when used with the Universal Turing Machine.

The third number (355,274) is also a Turing Machine Number of the same UN+1 Turing Machine. It was also constructed by inserting an "optional Leading zero" into the first Turing Machine Number in Binary Notation, but in a different place. The Universal Turing Machine does not work well when using this number. It just runs for a long time, and then crashes with the error message that the Excel file does not have enough columns to complete this run of the Universal Turing Machine.

I myself have not yet been able to figure out how the Universal Turing Machine really works, but examples such as Turing Machine Number 355,274 seem to tell me that it was not designed to handle Turing Machine Numbers with "optional leading zeros" in them. To me, that's perfectly all right. As long as we know that, then we know that we should only provide the Universal Turing Machine with the smallest Turing Machine Number for any Turing Machine, and it should work just fine.

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After working on the construction of this website for a few years, and after running the Universal Turing Machine on very, very many examples, I became convinced that the Universal Turing Machine will always work well, as long as the Turing Machine Number has no "optional leading zeros."

However, that is not the case.

When I was preparing a Help Video for one of the steps in the Notation Converter (which you can download from the home page of this website), I chose to construct a simple Turing Machine consisting of only three instructions. I was very surprised to find out that, even though the Turing Machine I constructed worked as expected, the Universal Turing Machine did not work as expected with its Turing Machine Number.

I knew that something was wrong somewhere. It occurred to me that there might be an error in the Universal Turing Machine, but I thought it was probably much more likely there was a problem with either of the two Excel files that I prepared for download from the main page of this website. I tried to see if I could find the error somewhere in the VBA code I wrote in these Excel files, but did not find anything wrong. I wished very much that I understood the logic that was used in the execution of the Universal Turing Machine, but I was not able to figure that out either.

While I was doing all this, I thought of something that I thought that I should dismiss as whimsical. Could Sir Roger Penrose have purposely planted a Paper Town in his Universal Turing Machine, and I just simply stumbled upon it?

The more I thought about it, the more it somehow appealed to me as very interesting. I seriously do not think there is a Paper Town in the Universal Turing Machine, but I thought I'd mention it in an FAQ anyway ---- just in case.

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In the reading above, I mentioned a Help Video. In case you're wondering how you can get to this Help Video yourselves so that you can watch it, here are the steps you should take:

Click on the picture in the main page of this website to download the Notation Converter Excel file (if you haven't done so already).

With the Notation Converter Excel file open to the Welcome sheet, click on the picture.

When it asks you "From which numerical notation will you be converting your number?" click on Machine State.

When it asks you "What Machine States shall we use?" click on I will provide my own Machine States.

When it asks you "How will you provide your Machine States?" click on I'd just like to type it in.

Then, in the Userform that pops up in the upper left corner of the screen, click on the box Help Video.

Then, just watch the video. Enjoy.