A Math Problem for You (1 Viewer)

pecorino · Apr 20, 2020

I read that Princeton professor and mathematician John Conway died last week (link) and thought the best way to honor him would be to get FBGs talking math. He was kind of like a Lebron of mathematicians, a man among boys.

You're probably familiar with graph paper. If not, this image may refresh your memory.

The graph paper shows grid lines drawn vertically and horizontally at whole number values, positive and negative (0, 1, 2, 3, 4, ... in the x-direction and 0, 1, 2, 3, 4,... in the y-direction) and where they meet are called lattice points (like (-1, 4) or (0, -3), or (99, 213)).

Question: Can you place a diagonal line on graph paper so that, even as the line extends forever in both directions, it never hits a lattice point? If you can, demonstrate how and if you cannot, prove that any diagonal line must hit a lattice point. Maybe put answers in spoiler boxes? Thanks and good luck.

Mjolnirs · Apr 20, 2020

Damn, I had not heard of his passing.

I'm not even going to try your question. But, I will say that I learned his doomsday algorithm for calculating the day of the week for any date. I've watched videos of him doing it, and his mental calculating speed is crazy.

BroadwayG · Apr 20, 2020

<spoiler>seems like any line with a slope of 1 or -1 and a non-integer intercept would qualify</spoiler>

SoBeDad · Apr 20, 2020

BroadwayG said:
<spoiler>seems like any line with a slope of 1 or -1 and a non-integer intercept would qualify</spoiler>

Seems like the obvious answer, so there must be more to it. BTW, he died in a NJ nursing home where he had been for almost a year.

pecorino · Apr 20, 2020

SoBeDad said:
Seems like the obvious answer, so there must be more to it. BTW, he died in a NJ nursing home where he had been for almost a year.

I did not know that detail. Lots of yucky stuff happening in NJ lately.

Good answer, BroadwayG. Now what if we require that the line goes through the origin (0, 0)?

bostonfred · Apr 20, 2020

I think the answer for that is any line with a slope that's an irrational number would never intersect a lattice point other than 0, 0. Because if it did, it wouldn't be irrational.

heckmanm · Apr 20, 2020

bostonfred said:
I think the answer for that is any line with a slope that's an irrational number would never intersect a lattice point other than 0, 0. Because if it did, it wouldn't be irrational.

And you could avoid (0,0) by using another (or the same) irrational number for the intercept. Example:

Y = eX + pi

BobbyLayne · Apr 20, 2020

7 to 1

heckmanm · Apr 20, 2020

heckmanm said:
And you could avoid (0,0) by using another (or the same) irrational number for the intercept. Example:

Y = eX + pi

Actually, just make the intercept irrational, no?

Y = X + pi

Bossman · Apr 20, 2020

heckmanm said:
And you could avoid (0,0) by using another (or the same) irrational number for the intercept. Example:

Y = eX + pi

this is what I was thinking.

but I'm a dope ... I'm sure there is more to it.

pecorino · Apr 20, 2020

heckmanm said:
Actually, just make the intercept irrational, no?

Y = X + pi

Yes, I think making both the slope and the intercept irrational is actually overkill and would be harder to prove it misses all lattice points but same idea.

pecorino · Apr 20, 2020

bostonfred said:
I think the answer for that is any line with a slope that's an irrational number would never intersect a lattice point other than 0, 0. Because if it did, it wouldn't be irrational.

Very good. Not sure how many folks know what an irrational number is.

pecorino · Apr 20, 2020

Conway dabbled with sequences. Or, I should say, he essentially wrote the encyclopedia on them in the 1970s. One sequence that caught his attention is shown below. Some of you may have seen this one (it is now well known) but what most people don't know is that Conway sought to find a pattern so as to say "the formula for term "n" is ______" and to create such a formula. That was impossible even for him, but he did determine some facts about the ratios of one term divided by the prior term. It's the kind of thing no one in their right mind would even consider working on. Conway worked on it and figured out how those ratios evolve.

1, 11, 21, 1211, 111221, 312211, ___________, ___________, .....

SoBeDad · Apr 20, 2020

pecorino said:
Yes, I think making both the slope and the intercept irrational is actually overkill and would be harder to prove it misses all lattice points but same idea.

What's rise over run for a line with irrational slope?

pecorino · Apr 20, 2020

SoBeDad said:
What's rise over run for a line with irrational slope?

For something like slope of square root of 2, you'd run 1 and rise the square root of 2.

TheWinz · Apr 20, 2020

Pie are squared?

SoBeDad · Apr 20, 2020

pecorino said:
For something like slope of square root of 2, you'd run 1 and rise the square root of 2.

So one could graph at most one point on a line with irrational slope?

pecorino · Apr 20, 2020

SoBeDad said:
So one could graph at most one point on a line with irrational slope?

At most one lattice point, correct.

Mister CIA · Apr 20, 2020

Four?

pecorino · Apr 20, 2020

Mister CIA said:
Four?

No but that fine gentleman did utter one of the all-time lines in cinema history, something about straightening out his Longfellow. Always makes me giggle.

wikkidpissah · Apr 20, 2020

the key to my success in life is that i was overcome by the desire to go to the bathroom for a smoke halfway thru the OP

pecorino · Apr 20, 2020

wikkidpissah said:
the key to my success in life is that i was overcome by the desire to go to the bathroom for a smoke halfway thru the OP

Words and numbers are not mutually exclusive. You don't know what you missed.

A Math Problem for You (1 Viewer)

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