Raising Squares Part 1

Logic Level 1

\LARGE \square^{\square^{\square^{\square}}}

You are given that the numbers 1,2,3,41,2,3,4 are to be filled in the square boxes as shown above (without repetition), forming an exponent towers? . Over all 4!=244!=24 possible arrangements, let SS be the minimal value that is achieved. How many arrangements of these numbers would produce the value of S?

Details and Assumptions

As an explicit example, a possible value of the resultant number is 2314=8\large 2^{3^{1^4}} = 8 .

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