2019/quals/mr_x.html

<p>
"There's nothing more important than x!", laughs Mr. X as he explains a Boolean expression involving a variable x to you and your classmates. 
He can't go 5 minutes teaching Boolean algebra without making at least one such "joke"...
</p>

<p>
In Mr. X's class, you've been learning about single-variable Boolean expressions, which are made up of the variable x (and its negation), Boolean constants (True/False), and binary Boolean operators. A valid expression is a string in one of the following two forms:
</p>

<p>
1) A single term, which is one of the following four characters:
<ul>
<li> "<code>x</code>": The variable x </li> 
<li> "<code>X</code>": The negation of the variable x </li> 
<li> "<code>0</code>": The constant False </li> 
<li> "<code>1</code>": The constant True </li> 
</ul>
</p>

<p>
2) A binary operator joining two valid expressions in the format "<code>([expression][operator][expression])</code>", with the operator being one of the following three characters:
<ul>
<li> "|": The OR operator (evaluating to True when at least one of its operands is True) </li> 
<li> "&": The AND operator (evaluating to True when both of its operands are True) </li> 
<li> "^": The XOR operator (evaluating to True when exactly one of its operands is True) </li> 
</ul>
</p>
    
<p>
For example, the following expressions are <strong>valid</strong>:
<ul>
<li> "<code>1</code>" </li>
<li> "<code>(x^0)</code>" </li>
<li> "<code>((X&0)|x)</code>" </li>
</ul>
</p>
  
<p>
While the following expressions are <strong>invalid</strong>:
<ul>
<li> "<code>(1)</code>" </li>
<li> "<code>x^0</code>" </li>
<li> "<code>(X&0|x)</code>" </li>
</ul>
</p>

<p>
An upcoming test will feature a valid expression <strong>E</strong> in the above format, which must be evaluated for a certain value of x. 
However, you've been getting tired of Mr. X and his lame jokes about the importance of x, 
so you're planning on hacking into his test files and changing the expression so as to make x irrelevant! 
In particular, you'd like to modify as few characters as possible in <strong>E</strong> such that it ends up still being a valid expression, 
but such that its overall value doesn't depend on the value of the variable x. You may only change characters in-place into different characters &mdash; you may not insert or delete characters.
</p>

<p>
For example, the expression "<code>(X|(0&x))</code>" evaluates to True if x is False, and False if x is True. 
If it were to be changed into "<code>((X&0)&1)</code>" (by modifying its 2nd, 3rd, 4th, 6th, 7th, and 8th characters), then it would evaluate to False regardless of x's value. 
Though, it's also possible to make its value independent of x by modifying fewer than 6 of its characters.
</p>

<p>
Given an expression <strong>E</strong>, what's the minimum number of characters which must be modified? It's possible that no characters may need to be modified at all.
</p>


<h3>Input</h3>

<p>
Input begins with an integer <strong>T</strong>, the number of tests.
For each test, there is a line containing the expression <strong>E</strong>.
</p>


<h3>Output</h3>

<p>
For the <em>i</em>th test, print a line containing "Case #<em>i</em>: "
followed by a single integer, the minimum number of characters to modify in <strong>E</strong> such that the result is a valid expression whose value doesn't depend on the value of x.
</p>


<h3>Constraints</h3>

<p>
1 &le; <strong>T</strong> &le; 500 <br />
1 &le; |<strong>E</strong>| &le; 300 <br />
</p>

<h3>Explanation of Sample</h3>

<p>
The first expression can, for example, be changed to "<code>1</code>" (and would then always evaluate to True).
</p>

<p>
The second expression can be left unchanged (as it always evaluates to False).
</p>

<p>
The third expression can be left unchanged (as it always evaluates to True).
</p>

<p>
The fourth expression can, for example, be changed to "<code>((0^(X&X))|x)</code>" (and would then always evaluate to True).
</p>