Not sure why my solution is wrong to 737Div 2 C. Probably my approach is wrong

Правка en1, от butterflies, 2021-08-10 08:02:39

Problem Link 737Div2 C

This is my solution. First, represent the $$$n$$$ numbers in binary. Each number has $$$k$$$ digits, where each digit can be $$$0$$$ or $$$1.$$$ Let $$$a_{i, j}$$$ represent the $$$j$$$th digit (from left to right) of $$$a_i$$$, where $$$a_i$$$ is represented in binary.

Now,

$$$a_1 \,\&\, a_2 \,\&\, a_3 \,\&\, \ldots \,\&\, a_n \ge a_1 \oplus a_2 \oplus a_3 \oplus \ldots \oplus a_n$$$

is true if for all $$$1 \le j \le k,$$$

$$$a_{1,j} \,\&\, a_{2,j} \,\&\, a_{3,j} \,\&\, \ldots \,\&\, a_{n,j} \ge a_{1,j} \oplus a_{2,j} \oplus a_{3,j} \oplus \ldots \oplus a_{n,j}.$$$

Let's consider the inequality for $$$j = 1,$$$ because others values of $$$j$$$ are similar.

$$$a_{1,1} \,\&\, a_{2,1} \,\&\, a_{3,1} \,\&\, \ldots \,\&\, a_{n,1} \ge a_{1,1} \oplus a_{2,1} \oplus a_{3,1} \oplus \ldots \oplus a_{n,1}.$$$

If more than one of $$$a_{i,1} = 0$$$ for $$$1 \le i \le n,$$$ then the left side of the above inequality is going to be zero. This requires the right side of the above inequality to be zero, so an even number of $$$a_{i,1}$$$ should be $$$1$$$.

Also, if all of $$$a_{i,1}=1,$$$ then that is another possible case. This only needs to be considered if $$$n$$$ is odd.

Now, using the above two paragraphs, there are

$$$\binom{n}{0} + \binom{n}{2} + \binom{n}{4} + \cdots + \binom{n}{n - n \mod 2} + n \mod 2 = \sum_{p = 0}^{\lfloor n/2 \rfloor} \binom{n}{2p} + n\mod 2$$$

ways to pick the $$$1$$$st digit for all $$$a_i.$$$

Using the Binomial Theorem on $$$(1+1)^n$$$ and $$$(1-1)^n,$$$ we can show that $$$\sum_{p=0}^{\lfloor n/2 \rfloor} \binom{n}{2p} = 2^{n - 1}.$$$

Thus, the number of ways to pick the $$$1$$$st digit for all $$$a_i$$$ is $$$2^{n - 1} + n\mod 2.$$$ Similarly, this is the number of ways to pick the $$$j$$$th digit for all $$$a_i.$$$

Since there are $$$k$$$ digits total in each of $$$a_i,$$$ the answer is $$$(2^{n - 1} + n \mod 2)^k.$$$

My code describes this approach, and I think it performs this approach correctly. Thus, I believe my approach to the problem is incorrect. Can anyone point out what is wrong?

I have also attached by code in case it is wrong.

Thanks for the help everyone!!

My code

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en1 Английский butterflies 2021-08-10 08:02:39 2979 Initial revision (published)