Turing Machine that halts iff the tape contains at least two '1's

Question

Contruct a Turing Machine that halts if and only if the tape it is on contains at least two '1's.

My attempt:

The gist of this turing machine is that it has to check all of the cells, and since it is infinite in both directions, one way of indexing it is a follows

Thus it has occurred to me to write a pseudocode for the TM's behaviour like this:

n:= 0           //number of '1's encountered
i:= 0           //index
while(true)
{
  if(n==2)
    halt
  for k=0 upto i
  {
    check()
    if(i even)
      move left
    if(i odd)
      move right
  }
  i++
}

Where check is the function defined as follows:

check()
{
if(current cell contains 1)
  n++
}

But this seems really hard to implement as a turing machine, especially since there are two variables involved. My guess is that I am overthinking and there is a much cleaner, nicer and elegant solution.

So my question is: What would be a better solution?

Answer 1

I managed to solve it in the end. If anyone is interested, I left a diagram of the TM here