# A small set numerical puzzle

Three,
Twenty-one,
and forty-five

start
this list
that has arrived.

---edit for a little more clarity---

Ninety-one
and one hundred and twenty.
Then we get to the doubly interesting number.

After which
comes two hundred and ten.
Three hundred and naught ends this list of wonder.
---end of edit---

Only
eight numbers
exist in this little set.

A
small pattern,
but one I hope you will get.

What
goes unnamed
here in my little list?

Why is
it interesting?
What gives it the twist?

If you
can solve my rhyme,
you'll feel rather quite peachy.

Can
you say
veni vidi vici?

My second finite list
only numbers but three.
Unfair on it's own,
a companion it'll be.

Start at forty nine;
then sixty plus four.
One hundred twenty one;
there is no more.
----end of edit----

Also twice as special
is one of these
Can you tell why

Edit for hints:

Hint 1

was going to be to point out the structure of my poem. But, since this hint is supposed to lead a reader to Dmihawk's conclusion, it hardly counts as a hint anymore.

Hint 2

There is a particular phrase that I hoped would stand out. While the phrase itself gives no direct hint as to the riddle, it does suggest (I hope) something to do with the set items.

----First Edit----

Edited to make it more obvious that there is a missing number in the first sequence.

original lines before edit:

Ninety-one
and one hundred and twenty
take us to the doubly interesting number.

Two hundred and ten
followed by (and ending with) three hundred
give us the final integers in this finite list of wonder.

----original lines before second edit to improve readability---- Four more than sixty,
and fifty minus one,
one hundred twenty one
and this list's done.

• Is the order in which the second list is presented significant, and are we intended to assume that the first list is in strictly ascending order (so 120<x<210)? Feb 7, 2019 at 16:10
• @Zomulgustar fair questions. Yes to the first list (and the unknown number's value), no to the second. I used that order strictly to make the stanzas in the second part all the same length. That is an area of the riddle I wouldn't mind editing for clarity &/or would be happy to have editing suggestions.
– Van
Feb 7, 2019 at 16:17
• Perhaps you could make that stanza ABAB by rhyming sixty-and-four with some variation off 'isn't any more'? Feb 7, 2019 at 16:23
• @Zomulgustar Let's see if that looks better
– Van
Feb 7, 2019 at 16:30

The missing number from the first set is

190

because the set consists of

those triangular numbers which when written as Roman numerals have length three.

The second set consists of

those squares which when written as Roman numerals have length four.

It is not yet apparent to me why the doubly-special ones are so. The obvious ways in this context to be twice as interesting would be

either to have another notable representation of length 3 (resp. 4), or (less plausibly) to be another kind of k-gonal number with some other representation of length k. Well, 190 has length 3 in our number system, but so do e.g. 120 and 300, so that's clearly not it. 190 is a hexagonal as well as a triangular number (as are half of all triangular numbers) but again so are 120 and 300. 190 is also the product of exactly three primes, but none of our squares is the product of exactly four primes.

Obviously

the layout of the lines gestures towards the triangularity/squareness of the numbers they describe

and

"veni vidi vici" is directing us toward Roman numerals.

• Excellent! I'm tempted to give you the green check, because you got the unknown number and what makes these sets finite, and given that I didn't really explain why the doubly interesting numbers would be doubly interesting. You are on the right track with your thinking. I'll see if I can think up a stanza that hints at the doubly interesting numbers without giving away the answers.
– Van
Feb 8, 2019 at 1:30
• If I'm on the right track then maybe 64 is the doubly-special number from the second set. Of course I have one chance in 3 of guessing that even with no correct ideas of what's going on... Feb 8, 2019 at 10:54
• After some time, I've realized I don't know that I'll be able to write a new stanza that doesn't make it obvious, hence Gareth gets the tick. (I was thinking of adding a fourth square stanza, if anyone wants to give it a try.) The doubly interesting numbers were so because there indices are also written as 3 or 4 character roman numerals. Looking back on it, maybe not that interesting.
– Van
Feb 19, 2019 at 12:29
• Ah! I don't think I'd ever have got that. Feb 19, 2019 at 12:34

Not sure if it's relevant, but...

The first sequence

are all triangular numbers

The second sequence

are all square numbers

• Right track. But why these finite subsets?
– Van
Feb 4, 2019 at 21:53