32814번 - Pascal Meets Boole 다국어

Many people are familiar with Pascal's Triangle, a triangular arrangement of integers named after the French mathematician and philosopher Blaise Pascal (1623--1662). If we number the rows of Pascal's Triangle $1, 2, 3, \ldots,$ starting from the top, then row $r$ contains $r$ elements, which we will number $1, 2, \ldots, r$ from left to right. The $1$^st and $r$^th elements in row $r$ are set equal to $1$, and for $r \geq 3$ and $1 < i < r$, element $i$ in row $r$ is the sum of elements $i-1$ and $i$ in row $r-1.$ More informally, each "non-edge" element is the sum of the two elements immediately above it. Figure J.1(a) depicts the first $8$ rows of Pascal's Triangle.

Figure J.1: (a) Pascal’s Triangle, (b) Pascal-Boole triangle for function $1000$

But what if we consider a rule other than standard addition for combining values? Since the edge elements are bits $1$'s), a natural option is to use any two-input Boolean function, named after the English mathematician and philosopher George Boole (1815--1864). For example, the Boolean function given by the following truth table generates the triangle depicted in Figure J.1(b) (where we also show the first $8$ rows). In this truth table, $x$ and $y$ correspond to elements $i-1$ and $i,$ respectively, in row $r-1$, and $f(x,y)$ is the resulting element $i$ in row $r$.

In general, if we label the bits in the rightmost column of any such truth table $b_{00}, b_{01}, b_{10}, b_{11}$ from top to bottom, then we can compactly represent a two-input Boolean function by the $4$-bit string $b_{00} b_{01} b_{10} b_{11}.$ So the example function above is represented by $1000$.

Your challenge is to answer two kinds of questions about "Pascal-Boole" triangles:

1. For a given Boolean function, $f,$ what is the bit in row $r$, position $i$?
2. For a given Boolean function, $f,$ how many $1$'s are there in the first $r$ rows?

The first line of input contains an integer, $n$ $(1 \leq n \leq 250),$ the number of test cases. This is followed by $n$ lines, each of which has one of two forms:

• $f$ B $r$ $i$
• $f$ N $r$

In both cases, f is a $4$-bit binary string representing a two-input Boolean function, and $r$ is an integer $(1 \leq r \leq 10^6).$ In the first case, $i$ is an integer $(1 \leq i \leq r).$

For a test case of the form $f$ B $r$ $i$, output a line containing the bit in row $r$, position $i$ of the Pascal-Boole triangle generated using $f$. For a test case of the form $f$ N $r$, output a line containing the number of $1$'s in the first $r$ rows of the Pascal-Boole triangle generated using $f$.