Oh no, I've deliberately obscured large portions of this ruler and I need to make sure these vegetables are whole numbers of inches long or my toddler will eat me instead: a #RealWorldMaths thread

As you might know, my daughter is both a very fussy eater and a superhero whose superpowers are an unbounded appetite and precisely eyeballing lengths.
Everything she eats needs to be a whole number of inches long (thanks for telling her about inches, grandad!)

When I stand up straight I'm exactly 6'5" tall.
I need to get these measurements exactly right, and I don't want to think about the alternative.

Only the 0, 1, 4 and 6 marks are visible. So I can definitely cut sticks that are 1, 4 and 6 inches long, like this:

But how can I measure 2, 3 or 5 inches?

Fun maths fact: 2 = 6 - 4. So to make sure something is 2 inches long, I just need to line it up with the 4 and 6 marks.

I can use some more Advanced Maths Theorems such as 3 = 4 - 1 and 5 = 6 - 1 to measure the two other lengths, like this:

Hooray, all the food is the right length and I will live to see another meal time!

But have you noticed that there are 4 × 3 ÷ 2 = 6 ways of picking two marks from 4? So this ruler gives me as many different lengths as it possibly can!

Rulers with this property - as many distinct lengths as possible for a given number of marks - were investigated by a few people, including Simon Sidon, Wallace Babcock, Sophie Piccard and Solomon Golomb, in the 1930s.

Ready for the punchline?

This is the best you can do!

There's no ruler with more than four whole-number marks that can measure every possible length up to the distance between the lowest and highest mark.

That's a lemma, a theorem, and a stone cold fact.

If you find that hard to believe, like I do, you can read a proof on Justin Colannino's page about Golomb rulers: cgm.cs.mcgill.ca/~athens/cs507/…
It's full of facts about all sorts of fruity variants on the setup, like circular rulers and modular rulers.

And if you were keen to get involved with the burgeoning field of Ruler Maths but are sad that it's been entirely resolved, take heart: there's still something we don't know and that you can help with!

If you're relaxed about measuring *every* length and just want a ruler that measures more different lengths than any other, nobody knows a formula for making one!

At the moment, we know the best rulers for up to 27 marks. After that it's all conjecture.

So you can either sit down and do some hard thinking and come up with a formula, or you can join the distributed effort to check ever bigger rulers using computers.

(One of these is easier than the other)

End of thread!