"All in" anti-Pascal triangle; (This could be a legendary weapon)

Ladies and Gentlemen, here it comes, from the deepest pit of hell, the LINE that divides mortals from Demigods;

A problem so hardcore, it's reputation better be unknown, so people don't quit before trying. Let me be clear, if you can solve this, you won't be a mortal anymore, you would be a DemiGod or a Goddess! Your soul would be near ultimate perfectness.

An anti-Pascal triangle is an equilateral triangular array of numbers such that, except for the numbers in the bottom row, each number is the absolute value of the difference of the two numbers immediately below it.

For example, the following array is an anti-Pascal triangle with four rows which contains every integer from 1 to 10.

                                                                           4
                                                                        2    6
                                                                    5     7     1
                                                                 8     3     10    9

Does there exist an anti-Pascal triangle with 2018 rows which contains every integer from 1 to 1 + 2 + · · · + 2018?

Side note: Note that the exact amount of int variables of the triangle is equal to the summation of every integer from 1 to n = amount of rows.

#Combinatorics

Easy Math Editor

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Comments

I used a computer program, and:

for $n = 3$ there are $4$ possible bottom rows: $(1, 3), (2, 3), (3, 1), (3, 2)$

for $n = 4$ there are $8$ possible bottom rows: $(6, 1, 10, 8), (6, 10, 1, 8), (8, 1, 10, 6), (8, 3, 10, 9), (8, 10, 1, 6), (8, 10, 3, 9), (9, 3, 10, 8), (9, 10, 3, 8)$

for $n = 5$ there are $2$ possible bottom rows: $(6, 14, 15, 3, 13), (13, 3, 15, 14, 6)$

Unfortunately, there were too many possibilities to test $n > 5$ .

David Vreken - 2 years, 10 months ago

Bro, You can have the anwser:

http://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMi81Lzk3MzFkYjRiOWFkZDI5NjhiZDdmYmU0NGI0MWNmOTQ1OGVhZGI1LnBkZg==&rn=dG9hZHMucGRm

Xizlon Tho - 2 years, 10 months ago

Oh, this is great! Thanks!

David Vreken - 2 years, 10 months ago

@David Vreken – No problem bro. And now, that I know you know how to program, later on, I will show you an interesting project I am working on; Because I need GRU neural networks to operate the markets with a unique tool that I have programmed.

Xizlon Tho - 2 years, 10 months ago

If you change all the even numbers to $0$ and all the odd numbers to $1$ , you can use a computer program to at least eliminate some larger triangles. (Two evens or two odds make an even, and one even and one odd make an odd, and we would require the whole triangle to have round( $\frac{n(n + 1}{4})$ odd numbers for $n$ rows.)

There are no possible even-odd bottom row combinations for $n = 6$ or $n = 14$ that will create triangle with round( $\frac{n(n + 1}{4})$ odd numbers, so there are no anti-Pascal triangles with $6$ or $14$ rows.

Unfortunately, this method did not eliminate $n = 22$ like I was hoping. I was able to test $n \leq 22$ before there were too many possibilities for my computer program, and $n = 6$ and $n = 14$ were the only $n$ -values that I could eliminate.

David Vreken - 2 years, 10 months ago

WOW, oh boy, If I were not with the flu, I would be deeply analyzing your approach; It looks very smart :D But, I do have the flu right now, :v

Xizlon Tho - 2 years, 10 months ago

I'm starting to recover; Mr. David Vreken.

Thanks to you, now I know there might be no solution for n=2018 rows, but, how can that be proven?

""A problem so hardcore, it's reputation better be unknown"" R/ Why?? Because this IS the hardest problem ever in the International Mathematical Olympiad, it´s from this year(2018) Olympiad; It was proposed by one of the fathers of the Cryptographic field, Martin Gardner.

That is why, I was being literal.

Xizlon Tho - 2 years, 10 months ago

Hello Mr. Agnijo Banerjee

https://www.imo-official.org/yearregteam.aspx?year=2018&code=UNK&language=es

I have been expecting you; I knew you will come here;

Do you want to see something amazing?

I have been developing a neural network that would theoretically allow me to generate money, predicting changes in currency and stock prices; Well, it's not as effective as your 3-book publication, but it's still entertaining.

Leaving that aside; I would like you to be my teacher when I have the ability to pay you; In return; we could investigate novel ways to help your brother; or whatever you want.

PD: I will show you, in time, with my effort, that I will be smarter than you :3

Xizlon Tho - 2 years, 9 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$