# MathBlocks Proof of Math --- Round 1 Questions
### Question 1
Let $S$ be the set of all possible 16x16 binary matrices, that is, whose elements are either 0 or 1. Calculate the number of elements of $S$ which have **at least one row or column** consisting of all ones.
Submit your answer as an integer. (Submissions in the format 1.2345e67 will be counted as incorrect)
### Question 2
Calculate (up to at least 6 significant figures) the probability of drawing two random elements from $S$ at the same time, both of which have **at least one row or column** consisting of all ones.
Submit your answer as a fraction in decimal or engineering notation.
### Question 3
The function $B: [0, 2^{256}-1] \to \{0,1\}^{256}$ converts a given positive integer less than $2^{256}$ to a zero-padded binary word of length 256. For example,
$$
B(67092511) = \underbrace{000\dots00011111111111100000000011111}_\text{256 digits}
$$
Furthermore, $M:\{0,1\}^{256}\to\{0,1\}^{16\times 16}$ casts such binary words into a 16x16 matrix:
$$
M(B(67092511)) =
\begin{bmatrix}
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&1&1&1&1&1&1&1&1&1&1\\
1&1&0&0&0&0&0&0&0&0&0&1&1&1&1&1\\
\end{bmatrix}
$$
Finally, we denote by $H(\cdot)$ the [SHA-256 hash function](https://en.wikipedia.org/wiki/SHA-2), which outputs 256 bit hash values.
Find a positive integer $n < 2^{256}$ such that $M(B(n))$ and $M(H(B(n)))$ both have **at least one row or column** consisting of all ones.
Example of such an integer: 625544474832775601147535878970336692745258958953363790887354131949979176970.
Submit another one for the answer. If you submit this one, it will not count it as a correct answer.
**Clarification:** Integers are encoded directly as their binary values, big endian, and zero padded as shown above. The data bytes that come from the encoding is then given as input to the hash function. Do **NOT** use character encodings of a bitstring as input, e.g. `"11010101"`'s UTF-8 encoding.
### Submissions
Submit your answers through the [Google Form by clicking here](https://docs.google.com/forms/d/e/1FAIpQLSfQXNLzUU6jQQePmZeOkqpqdDrYqPGzxB7ul5NLWsyqXMcCKw/viewform?usp=sf_link).
### Prizes and rules
- Contest starts on Feb 5, 2022 14:00 UTC. Submission window is 48 hours.
- To get the prize, you need to answer all 3 questions correctly.
- Everyone who submits a 100% correct answer in the first 48 hours will get POAPs.
- You need to be in our Discord to receive the prizes. [Click here to join](https://mathblocks.io/discord).
Prizes are:
* 1st correct submission: 0.15 ETH (450 USD)
* 2nd correct submission: 0.1 ETH (300 USD)
* 3rd correct submission: 0.06 ETH (180 USD)
* 4th correct submission: 0.05 ETH (150 USD)
* 5th correct submission: 0.04 ETH (120 USD)
Total: 0.4 ETH (1200 USD)
### About MathBlocks and Proof of Math
MathBlocks is an online community whose goal is to facilitate the creation of open educational resources. Proof of Math is a weekly math competition with cash prizes for those who like a good challenge. We additionally have daily challenges which are meant to be easier.
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