Seven primes?

Number Theory Level pending

Can you fill each box of a $3 \times 3$ grid with 9 (not necessarily distinct) digits such that every column (read from top to bottom) and every row (read from left to right) and the diagonal read from the top left to the bottom right represent seven (not necessarily distinct) three digit prime numbers?

Yes No

3 solutions

Michael Mendrin
Sep 15, 2018

Or this, where all of the primes are distinct, including both diagonals, and all primes are still prime when read in reverse.

1	1	3
7	5	1
9	7	1

Cool answer! I was just wondering if one such solution could be obtained.. 😃

Geoff Pilling - 2 years, 8 months ago

I just updated my solution

Michael Mendrin - 2 years, 8 months ago

Nice. That solves the "8 primes" question, then. :)

Brian Charlesworth - 2 years, 8 months ago

Can you find a 8-prime-magic square (3x3) that you can read from any way (e.g. not only from top to bottom, but also from bottom to top)? (probably called 16-prime magic square)

X X - 2 years, 8 months ago

You would be limited to using the 28 3-digit emirps in creating a "16 prime square", (actually only 14 3-digit sequences, as each must serve double-duty), so we should be able to check whether this is possible fairly quickly.....

Brian Charlesworth - 2 years, 8 months ago

Thank you!

X X - 2 years, 8 months ago

Unfortunately, not with all primes distinct from each other (at least not so far)

This is probably the best that can be done, where the primes are distinct if read one way, such as normally from left to right, and from top to bottom, and distinct again if read the opposite way.

1	1	3
7	5	1
9	7	1

Michael Mendrin - 2 years, 8 months ago

For all the primes to be distinct there would have to be 8 distinct 3-digit emirps with the same middle digit, which is not the case, (each of 0,3,4,5,6 are the middle digits of 4 different emirps).

Brian Charlesworth - 2 years, 8 months ago

@Brian Charlesworth – Ah shux you are right... I was looking forward to to seeing the grand daddy of all 3x3 prime boxes! 😃

Geoff Pilling - 2 years, 8 months ago

@Geoff Pilling – Although I wonder if we could do this for a 4x4 grid? Have 20 distinct primes filling all rows columns and diagonals (both ways)...

Geoff Pilling - 2 years, 8 months ago

@Geoff Pilling – I suspect that will be possible, as there are more primes to choose from. While the outside rows and columns can only have the digits 1,3,7,9, there are 10 4-digit emirp pairs that meet that requirement, and then there are over 100 emirp pairs to choose from for the interior rows, columns and diagonals. So the odds look good. :)

Brian Charlesworth - 2 years, 8 months ago

Thank you!

X X - 2 years, 8 months ago

Nice solution... I should have made this the problem!

Geoff Pilling - 2 years, 8 months ago

A Former Brilliant Member
Oct 1, 2018

Or this, where all of the primes are distinct and the other diagonal is the magic number.

7,6,9

5,5,7

1,3,7

Geoff Pilling
Sep 15, 2018

Yes. As follows:

2	2	3
2	2	3
3	3	7

Interesting problem. I found the word "represent" a bit vague; perhaps if you had something like "every column (read from top to bottom) and every row (read from left to right) and the diagonal read from the top left to the bottom right" then the context of "represent" would be more clear. Just a thought. :)

Anyway, since 323 = 17 x 19 the right diagonal isn't prime as well. I wonder if there is a "prime magic square" in which this diagonal "represents" a prime as well? Or if there is a solution to your question where all the digits are distinct?

Brian Charlesworth - 2 years, 8 months ago

Good suggestion!

I've updated the question accordingly... 😃

Geoff Pilling - 2 years, 8 months ago

I don't believe we can have every digit be distinct since all 3 digit primes end in 1 3 7 ir 9, and we need 5 such digits... 🤔

Geoff Pilling - 2 years, 8 months ago

Ah, yes, good point; so the best we could do is 8 distinct digits with one repeat. This will be a fun problem to play around with, (including finding a square with 8 primes). :)

Brian Charlesworth - 2 years, 8 months ago

@Brian Charlesworth – Hahaha... Yup! I was just thinking about the 8 prime problem...

Geoff Pilling - 2 years, 8 months ago

Seven primes?

3 solutions

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