is "up to" the same as "less than" or "not greater"

[laune]

This function word occurs in several problems. Usually it's obvious that it means "less or equal". In 173, it isn't so obvious as many other problems have either "less than" or "not greater than" some limit.

I'm not sure whether native speakers have no doubts. To cite Merriam.Webster: "...sank up to his knees in the mud", which, I think, excludes the knees.

[uws8505]

I'm not a native speaker, but I thought "up to" as "not greater than" and got a correct answer.

[daniel.is.fischer]

I'm not a native speaker either, but my impression is that (at least in british usage) the primary meaning includes the limit.

[JohnMorris]

I am a native speaker. After playing a few examples in my mind, "up to" generally does include the limit, but there can be ambiguity sometimes. For example, colloquially I might say "Up to today, I have never seen a blue baboon". That suggests I have seen one today, so the "up to" there does not include the limit, "today".

I would suggest avoiding its use when clarity and precision are needed, especially for text that will be read by non-native speakers.

[daniel.is.fischer]

Can anybody suggest a few unambiguous alternatives to "not exceeding" or "not greater than"?
You know, variatio delectat.

[laune]

Can anybody suggest a few unambiguous alternatives to "not exceeding" or "not greater than"?
You know, variatio delectat.

Here are some phrases which I'd consider unambiguous: not beyond x, below or equal x, not in excess of x, up to and including x, not more than x.

And then there is always "not closer to positive infinity than x"

[JohnMorris]

Can anybody suggest a few unambiguous alternatives to "not exceeding" or "not greater than"?
You know, variatio delectat.

Tricky...

The original question referred to Problem 173 (View Problem), and the wording there is:

Using up to one million tiles...

Some alternatives would be

Using no more than one million tiles...

or

Using at most one million tiles...

or

Using one million tiles or fewer...

In this case the preceding sentence offers some clarification:

With one-hundred tiles, and not necessarily using all of the tiles at one time...

so you could simply repeat that form:

With one million tiles, and not necessarily using all of the tiles at one time...

The repetition is admittedly ugly.

[rayfil]

The repetition is admittedly ugly.

This may be true when writing a fiction novel. However, repetition is beautiful when writing anything technical. In such cases, it should even be considered as an absolute necessity to avoid any ambiguity.

Having done translations of highly technical material in the past, there was nothing worse than having an author writing the original material as if it was a novel, using different terminology for exactly same item!!!

As for the current subject, the meaning of "up to" may often not need to be defined further according to the goal to be achieved, such as when the upper limit would be impossible to use (example: primes up to 1000000).

A recent practice has been to use the ≤ or < symbols to specify the limits unambiguously.

[LarryC]

Good arguments... I personally prefer ≤ or < as you say.