Answer in 5 . 4 . 3 . 2 . 1 .....

[kriesel]

40 = 5*(4+3+2-1)
41 = 5*(4+3+2-1) + 1
39 = 5*(4+3+2-1) - 1

"using only the digits 5, 4, 3, 2, and 1, in that order." (No other digits are allowed, and they must appear in the specified order)
"You may not change the order of the numbers"
It does NOT state the digits must only appear once per value!

But this is an unsatisfying solution. It leads to a general case where, for any arbitrarily large number > 14, one can simply pile on the needed number of ones to be summed after a leading sequence such as
5+4+3+2

(Or for that matter: Must they all 5 appear at least once? Is the number base required to be ten?)
In base 30, 39 (base 30) = 99 (base 10)= 5*4*(3+2) - 1

15=55+4+3+2+1 has a typo; 15 = 5+4+3+2+1.
There are more results with multiple solutions, eg 31 = 5*4*3/2 + 1

[retina]

Originally Posted by kriesel

It does NOT state the digits must only appear once per value!

It also doesn't say we can't invoke the magic wizard spell MakusUppumShittius to give any answer we desire. So let's use that and solve all the puzzles.

Such interpretations of it-doesn't-say-X-so-let's-use-X render all puzzles useless and boring.

[kriesel]

Originally Posted by retina

It also doesn't say we can't invoke the magic wizard spell MakusUppumShittius to give any answer we desire. So let's use that and solve all the puzzles.

Originally Posted by R.D. Silverman

This is very clear. It specifies the operations that may be used.

The allowed operations do not include spell casting, nor do my suggested solutions.

If the puzzle was correctly written, specifying too completely would provide too obvious a hint:
Use only the digits 5, 4, 3, 2, and 1, in that order, at least once each.
Identifying unstated and mistaken assumptions and putting them aside is what may separate the student that gets full credit from the ones that get an A grade while missing some points.
39 = (55 + 44 ) / 3 + 3 + 2 + 1 uses only the digits 5, 4, 3, 2, and 1, in that order, but some more than once, and the only operators in the expression are ()/+.

On the other hand, having a set of rules which make the assigned task impossible to complete is an abuse of authority:
Use only the numbers 5, 4, 3, 2, and 1, in that order, exactly once each.
Or if the puzzle is fair, the appearance of no solution is a sign of an undetected incorrect assumption about the rules.

[S485122]

Originally Posted by kriesel

...
If the puzzle was correctly written, specifying too completely would provide too obvious a hint:
Use only the digits 5, 4, 3, 2, and 1, in that order, at least once each.
...

Perhaps the puzzle was adequately written but not sufficiently read ?

Take the digits 5, 4, 3, 2 and 1, in that order. Using those digits and the four arithmetic signs — plus, minus, times and divided by — you can get 1 with the sequence 5 - 4 + 3 - 2 - 1. You can get 2 with the sequence (5 - 4 + 3 - 2) x 1.
The question is ... how many numbers from 1 to 40 can you get using the digits 5, 4, 3, 2, and 1 in that order along with the four arithmetic signs?
You can group digits with parentheses, as in the example. There are no tricks to this, though. It's a straightforward puzzle. How many numbers from 1 to 40 can you get — and, specifically, what number or numbers can you not get?

In both posts where you refer to the condition in the first sentence and the second line of the puzzle, you add the word "only" that changes the puzzle, indeed it opens the door for using some elements of the set more than once. But the puzzle states to use a set that is enumerated (and no element of that set is repeated).

The rest of your post is about another situation altogether : did you note that the questions leaves a possibility that a number can't be represented, and then goes as far as asking which they or it might be ? The solution to the puzzle, its point, is to find out that 39 can't be represented.

But I agree the puzzle is not stated in a formal mathematical way : it should speak of numbers and not digits ; it is not digits that one can group with parentheses, but operations, otherwise it opens the way to concatenation ; it doesn't specify the base ; perhaps one should specify the set and define the operations used ; ... It should be possible to state this puzzle in such a way that only a very few can understand it ;-)

[R.D. Silverman]

Originally Posted by S485122

it doesn't specify the base

Ah, but it did when it said digits! It did not say, e.g. bits, or octets etc.

[slandrum]

Originally Posted by R.D. Silverman

"You may use addition, subtraction, multiplication, division, and parentheses as many times as needed"

This is very clear. It specifies the operations that may be used. Concatenation is not among them.
Otherwise, one could use your "exclusion" argument to say that exponentiation and other operators
(e.g. xor, and, etc) might also be used because they were not excluded.

It doesn't say anything at all about operators. Parentheses are not operators either. You are reading more into the rules than are stated.

[retina]

Originally Posted by R.D. Silverman

Ah, but it did when it said digits!

Curious. I would interpret digit to be valid for any base.

Binary digit
Octal digit
Decimal digit
etc.

[Dr Sardonicus]

Originally Posted by retina

Curious. I would interpret digit to be valid for any base.

Binary digit
Octal digit
Decimal digit
etc.

In the present instance, the digits 1, 2, 3, 4, and 5 indicate a base of at least six. Since the numbers (in base ten) 1 to 38 can be represented under the rules, but 39 (base ten) can not, and 40 base nine is 36 base ten, we can say that ten is the smallest base for which there is a number less than 40 which is not representable.

[LaurV]

Originally Posted by retina

Curious. I would interpret digit to be valid for any base.
Binary digit
Octal digit
Decimal digit
etc.

+1. The "bit" itself is just a short for "binary digit". I had this argument with RDS some time ago on this forum It turned ugly.
However, I agree with RDS that concatenation should not be used, ESPECIALLY because the puzzle says "digits", putting 4 and 3 together to get 43 is, in my mind, clearly, an operation which is called "digit concatenation".

[Dr Sardonicus]

Originally Posted by LaurV

<snip>
However, I agree with RDS that concatenation should not be used, ESPECIALLY because the puzzle says "digits", putting 4 and 3 together to get 43 is, in my mind, clearly, an operation which is called "digit concatenation".

The original puzzle (linked to in the IP of this thread) says

Next week's challenge: This is a two-week challenge. Take the digits 5, 4, 3, 2 and 1, in that order. Using those digits and the four arithmetic signs — plus, minus, times and divided by — you can get 1 with the sequence 5 - 4 + 3 - 2 - 1. You can get 2 with the sequence (5 - 4 + 3 - 2) x 1.

The question is ... how many numbers from 1 to 40 can you get using the digits 5, 4, 3, 2, and 1 in that order along with the four arithmetic signs?

You can group digits with parentheses, as in the example.

And there you have it. Plus, minus, times, divided by, and grouping with parentheses. That's it.

[slandrum]

Originally Posted by slandrum

39 = -5 + 43 + 2 - 1

It says to use the digits in order, not the numbers 5,4,3,2,1. My answer above uses the digits in order along with + and -. It clearly fits within the stated instructions. Problems of this sort are often stated like this so that in order to answer them, you have to let go of an automatic assumption.

My answer above is not the only answer either. 5 * 4 * 3 - 21 also works.