## Sunday, October 16, 2016

### NPR Sunday Puzzle (Oct 16, 2016): Five, Four, Three, Two, One, Blast-off!

NPR Sunday Puzzle (Oct 16, 2016): Five, Four, Three, Two, One, Blast-off!:
Q: 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. 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? Will Shortz will reveal his solution in two weeks.
Enjoy the two-week math challenge. Feel free to post *how many* numbers you can create, but just not specifics on which ones and with what expressions.

Edit: A bit of a trick here. You can create 39 numbers, but not 39. :)
A: Without resorting to tricks, you can write expressions for all but number 39.

1. Given it is a two-week challenge and math oriented, I'm going soften the rules for this puzzle.

You may provide a count of how many of the 40 numbers you can create but just not specifics on which ones and using which expressions. Let's see if anyone can out do Mr. Shortz.

You have two weeks to come up with your answer and to submit it to NPR via the link above. Thank you.

2. I am guessing fractions are ok, since "divided by" is included (?).

1. Ah, I see. 5, 4, 3, 2, 1 must remain in that order.

3. "Almost 900" correct answers this week, in case anyone cares.

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1. Hmm... It's going to take people awhile to figure out their answers and I think giving people a gauge on their progress compared to others could be helpful. I suppose if someone gets to all 40, then we know which numbers are/aren't achievable but otherwise I don't think the number gives to much away. Thoughts from others?

2. I agree with Blaine, this isn't a puzzle with a definite solution, it's a competitive puzzle for the highest score, and knowing where others are doesn't give you any answers.

Is it too much to say the 30's will be the hardest to achieve?

3. OK. I was accidentally logged in under my wife's account above. I withdraw my reservations.

5. David, ViolinTeddy, sdb and other Blainesvillians -- all y'all OK in the northwest wild weather?

1. All is good here in Seattle. It sounds like the Portland got more wind than Seattle. My run today weather-wise was uneventful. Thanks for asking.

I did get to hear Yo-Yo Ma on Friday night and Itzhak Perlman today with the Seattle Symphony, unaffected by the weather.

6. I have ALL but one of the 1-40 solutions.

1. Hmm. I get 38.

2. I'm with Ron, and I think (for a change) we have the same pronouncement.

3. My bad (programming error). I agree with Ron and ecoarchitect.

4. I have a way to get all 40, though some may consider it cheating.

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6. I think that's giving too much away.

7. That's where I was headed, also. It seems to fit the puzzle constraints and I don't think it falls into the "trick" category.

8. I have yielded to your objection, ecoarchitect, but see the discussion below.

9. Can't control all the children (or voters).

7. I won’t be playing this time. I don’t enjoy the type of puzzle where at any given point you don’t really know if you’re through or not. Should you devote more time to it (or an element of it) that you haven’t solved yet, or should you decide that it’s insolvable and move on? It becomes a time management issue... I’d rather take a walk :)

1. This is one of the famous "before I got out of bed" challenges.

2. With your dry humour, Mendo Jim, are you sure you don't live in Arizona?

3. It is nice and wet this weekend in Mendoland.
By "before I got out of bed," I meant when I decided to take two weeks off. I hope I remember to come back.

4. Good puzzle to solve during a long run.

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9. I'd like to say how many I got, but I'm not sure how to do so without saying what number(s?) I can't get.

10. I have a hunch it's XL also but I have XXXIX so far.

1. If this is a reference to Will's clue, and your success count, I agree.

2. I have them all now and, for the two most difficult numbers, I have submitted 2 alternative answers for each.

3. None of my answers use combined or additional numbers.

11. L nzuc zlh dvnqpx ldv jzyy ...

1. Technically the way the puzzle is worded, I agree with you. But I would hope Will also intends for us to submit the way we got each number.

2. That sounds like an awful lot of work for the hapless NPR intern (while Will frolics in the cafes of Bratislava).

3. According to the World Puzzle Federation site, Senec, about 30 km from Bratislava.

4. Czech and mate....

5. A few points. My impression is that math-based puzzles tend to garner fewer responses to begin with. And many of the responses can be discarded based on simpler criteria. Lastly, I would not at all be surprised if Will has an automated method for checking puzzle responses.

6. They really only have to check the one they pick and if that's not ok, to pick another until one that's correct is chosen.

I've noticed some weeks they say approximately how many correct answers they got and other weeks just how many entries. I wonder if that is related to how hard it is to check or if that is just random.

7. Sometimes WS also announces alternate answers received, so it's unlikely they stop at the first "correct" submission. No fun being an intern.

More so with this one, the consensus with this group is there is 1 number that can't be achieved with our understanding of the rules, but Ron (and perhaps other tricksters) have answers that may be acceptable. Or at least worth mentioning on air.

8. And just to anticipate SDB, do interns stay at an internment camp?

9. Only if they get caught in an intern net.

10. Bill Clinton did have to deal with internal affairs.

11. I think he was SPOT on when he adDRESSed that issue as it became internecine.

12. Locker room talk

13. . . .from an old locker room

14. Al Gore was immune from sexual harassment because he had a locked box.

15. I agree with WW, "internal affairs" was as far as I was going to go.

Big up side if Trump wins: lots of jokes to tell. We can tap them in Morse Code on our cell walls.

16. Eco:
You foolish optimist! Trump won't jail us when he has walls to stand us up against. He doesn't waste time with his short (not fingers this time) attention span.

17. Unless he assigns this detail to Pence!

18. Oh please! This is not the time to be Pensive.

19. SDB: I like it!
I'll probably watch tomorrow's debate but not looking forward to it. We might see the equivalent of a WWF match or a Jerry Springer show. :(

20. SDB: I like it!
I'll probably watch tomorrow's debate but not looking forward to it. We might see the equivalent of a WWF match or a Jerry Springer show. :(

21. I on the other hand am postponing the opera in order to watch it. You never know, they might even mention the issues. Chris Wallace may even feel compelled to do his job, but I doubt it.

22. Now who's the foolish optimist?

23. The idiot undecided voters who are expecting either candidate to act differently. It's just like a fool looking at a square peg and hoping it will suddenly become round.

24. Most folks I know are undecided, but I don't think they're idiots. Do they hold their nose (and stomach) and vote for HRC, or feel good about voting for Dr. Jill? An easier decision for those in solid blue states like CA or WA, harder for folks in VA or PA.

But I live in a world where no adult would want to vote for the short-fingered vulgarian.

25.  I live in an area where we border two states with key senatorial and congressional races plus a governor's race to boot. I can't stand watching any local TV programs unless I pre-record it. Almost every other commercial is political in nature. My gosh, if someone hasn't made up their mind by now...well, I'll just say that , around here, they must be in a very small voting minority. I can't wait for November 9th!!

26. My post above is referring to the numerous undecided voters I heard interviewed this morning on NPR who were asked what they would be hoping to see at the debate this evening. They all said either they were hoping to see Trump ditch the ego and act presidential or Clinton to become warm and fuzzy. This is idiotic! To expect an older adult to suddenly become a different person than he/she has always been makes no sense at all.

27. Ha, ha, ha, ha, ha. I have not seen a single political commercial this time around. I gave up on watching TV years ago and don't miss it one bit. I do sometimes watch 60 minutes, and occasionally hear about something worthwhile on PBS, but other than that and the debates, which I only watch for entertainment purposes, I only use the TV for watching DVDs.

28. SDB - I envy you! Just think of all the "gutter cover" commercials you are missing. I'm also stuck FFWD'ing through all the Humira prescription commercials, too. I wish I had "1/40th" of the advertising budget that is spent promoting that one!

29. SDB - I envy you! Just think of all the "gutter cover" commercials you are missing. I'm also stuck FFWD'ing through all the Humira prescription commercials, too. I wish I had "1/40th" of the advertising budget that is spent promoting that one!

30. I don't even know what Humira is, I am happy to say. I listen every day to NPR and even their so called non-commercials drive me nuts. Lumosity and the one about increasing your curiosity quotient are the worst, and then most of the others are a barrage of tag lines that seem to be getting longer and longer lately.

Speaking of missing ads, how many times have you watched a local TV news broadcast and after they finish with a short piece you realize it was so poorly done that you don't even understand what you just watched? Years ago when I did watch some TV news (oxymoron) I had that experience numerous times.

31. Ha, I agree with sdb about not knowing what "Humira" is. When you wrote that initially, 68Charger, I thought perhaps you were referring to Huma Abedin.

32. Humira is an injected medication to treat rheumatoid arthritis, inflammatory bowel disease, and similar autoimmune disorders. It's a monoclonal antibody that binds to tumor necrosis factor alpha. Usually prescribed by a rheumatologist. Being who I am, I'm struck by the fact that one of the rheumatologists at our hospital is named Humaira.

33. WW: Ha! Ha! No, Humira is that heavily advertised RX for arthritis, Crohns disease and Psoriasis. They just play those ads like crazy.
Well, almost time for the Debacle!

34. Thanks, jan and 68Charger. Strange about the rheumatologist Humaira!

What did y'all think of tonight's debate?

DT's refusal to say he will concede if he loses was truly disturbing.

And speaking of Huma Abedin, she sure tried hard to tear Hillary away from the crowd after the event was over.

35. I thought it very revealing where the Groper in Chief candidate repeated his assertion that no one has more respect for women than he; and then soon after he interrupted her saying, "She's a nasty woman."

36. Agreed, sdb. He simply cannot see women as anything other than someone to grope.

I did think the candidates spoke more about the issues than the other debates. That was a welcome change.

37. I thought Hillary won overall. I like both your comments but I wasn't looking for Huma and didn't see her hurrying up HC. I also think that it didn't really change anyone's mind.
Trump did validate Alec Baldwin's SNL characterization, though!
I am glad these debates are finally over.
CNN has a panel discussing all this right now and it is just a big argument where no one is going to back down one inch.
Hurry up 11/9/16!!

38. What a bizzare election cycle!!

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12. Do the directions mean that you must use an arithmetic sign between each of the numbers or can you use numbers like 54, 43, 32, and 21? It is not clear to me with the way the directions are worded.

1. How's this logic: Using 2-digit numbers like that, I was able to form all 40 numbers from 1 to 40. The wording of the puzzle suggests to me that there is at least one number you can't get. Therefore, we must not be allowed to use 2-digit numbers. Plus, this seems to be a "trick", which Will assured us wasn't needed.

2. Trick or treat! A two-week puzzle!

13. I have 2 questions: 1. Do answers that are fractions count? Do you think the answer can have a decimal as part of it? 2. Can you combine digits as jutchnbev suggested? Will did not specify that answers had to be whole numbers and he did not suggest that the digets could not be combined.

1. I do think it has to be whole numbers 1-40. Otherwise, using decimals or fractions there could be an infinite number of numbers from 1 to 40.

As to combining numbers, as in 54, 43, 32, and 21, it seems to fit the puzzle constraints but makes the answer very straightforward, as jan noted.

Maybe Will will weigh in from Slovakia. . .

14. I have all forty, given that you can use 54, 43, 32, and 21. I won't try to get it without using two-digit numbers, because I think this fits the requirements, and I do not think it is a trick. ---Rob

1. Given that Will asked "specifically, what number or numbers can you not get?", I really don't think that saying you can get them all by using 2-digit numbers is going to cut it.

15. So, those of you avoiding double digits: How many do you have so far? ---Rob

1. All but one.

2. Surely if you are down to one, you have the answer Will intended.

3. Do you suppose we get extra credit for numbers we find with more than one, not-tricky solution?

4. I have all but one. I used single digits, only the 4 mathematical signs plus parentheses, and no tricks - just straight Algebra I from long ago. I am stymied on the last one. Any use in thinking there are 40? (38 really, not counting the examples given for 1 and 2.) I'm going to poke at the remaining one for a bit. More power to Ron if he ran the table.

5. GB, if you're not using any tricks then you're not going to find the one that remains.

6. I am also at all but one.

I don't think we need to show our work. I believe an acceptable answer is in the form of "I can get xx of 40. The only number(s) I can't get is (are) ..."

7. Long time lurker. I sat here for about an hour and was able to get all but one, as well. no tricks, no double digits.

I was able to get 41 though.

8. Surely if you are down to one, you have the answer Will intended.

16. Does this puzzle require showing the calculations?

17. This puzzle may be a sign of the times, but it also has lots of pluses and minuses too, for which I do not wish to spend my time divided between solving it and contemplating all the exciting solutions Donald Trump has in store to improve our country. Besides, I prefer to leave calculus to my dentist to deal with.

The storm was no big deal except for those who lost power when a tree took out a power line. My trees still seem to have the same amount of leaves remaining.

18. Here's another question. Can you use 2 to indicate squaring of 3? Also, do we need to show how we got the 40 numbers. If we think we need to show our work, it will probably take me the two weeks to put this in the answer submit form.
btw, if I use adjacent numbers like 32 or 21, I can get all 40. If not, I cannot get one. I can get that one if I use a 2 as a square. I can also get there if I use other arithmetic symbols like ! Will did specify "the four arithmetic signs" but then went on to talk about parenthesis. Any thoughts ? Thanks!

1. I don't think Will intended to include exponentiation.

19. In addition to using two digit numbers, does anyone use a leading minus sign on the 5 to make -5? He said that there were no tricks but considering that this is where many of us went immediately, would that realty be considered a trick?

20. Using 2 digit numbers (54, 43, etc.) seems like it would make this puzzle really easy, and is a bit of a trick, probably a "7" on the upside-down alarm clock scale. I don't think Will intended that we could use it.

I am now intrigued (and devastated) that Ron wrote he could get all 40 numbers without using that trick. I remain stuck at 39. Back to the drawing board - no computer programs to figure this out for me.

1. eco, surely not truly devastated?

2. I don't think he's serious, either. Or Shirley.

3. ;-)

De vast state conspiracy?

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7. Eco, don't break your wrist. I can definitively state that there is no way to get the remaining number in the traditional manner, and the only way I've found so far is by allowing the removal of spaces. I haven't looked at other tricks (such as exponentiation), since that seems like it's going too far. Ron stated "none of my answers use combined or additional numbers," which is not the same thing as saying he got there without using some other trick.

Jan, it went over my head that this information hadn't already been openly discussed. I guess we're considering that a secret?

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9. Ron, as Jan noted, I don't think we should reveal the most difficult answer, best to delete your comment.

10. WW: Would you believe I left out the space, it should have been "Deva stated"? Many Hindu prayers have 40 stanzas, and like DT "I love Hindu", even though that makes no grammatical sense. But Hinduism has a tremendous number of holy days, and if we could convert that to holidays, I'm in! You don't have to believe that.

I wonder which "trick" Ron is using, or if he is following what many think are the rules: single digits only (though not stated by WS), +, -, * and / only, with as many ()'s as we want, and with no squaring or cubing or whatever.

11. I am using ÷ instead of / and x instead of *.

12. Yeah, I just can't figure out where the ÷ is on my keyboard.

13. I found it oddly satisfying to crank through, with pen and paper, the all-but-one answer.

Yet, I don't especially feel drawn to typing them all out. And I'm sure the NPR intern doesn't want all that...The journey not the destination and all that.

Destination Senec. . . Deva stating. . .

21. I'm staying out of this. I was never that good at math. Besides, I prefer word puzzles to number puzzles.

22. Having decided to forego the fun of doing this puzzle, I probably shouldn't comment, but I have never let such considerations stop me before.

If I were a puzzlemaster, I would spend a few minutes thinking about possible inconsistencies, ambiguities and lack of clarity before promulgating a puzzle, especially a two-week one.
But such considerations haven't stopped Will in the past either.

1. I agree, unless the ambiguity doesn't change the answer. I started using combined numbers, but was able to replace them with equations using non-combined numbers. Can't wait to see Ron's 40.

23. Aside from being a nice, even number, why did Will stop at 40? What's the next number above 40 that can't be made?

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2. I can't get 42 or 43, and 47-49 elude me.

3. 42, 43, and 48 require some combination of the aforementioned tricks (space removal and/or negative before the 5), but 47 and 49 can be found through standard means.

4. The next number above 40 that can't be made is 43, though it can be made by prepending a negative to the 5 and removing a space. Allowing those sorts of tricks, the only numbers below 100 that can't be made are 66, 68, 73, 74, 79, 88, 89, 93, 94, and 98.

I deleted my earlier comment because I had previously overlooked ways to construct the following numbers: 67, 76, 77, 84, 96, 97.

24. Blaine, how can one contact you? Thanks.

25. For extra fun over the next two weeks, calculate the following:

A) The largest prime number
B) The largest Mersenne prime
C) The primes greater than 100 (eleven total, including the above)
D) The number where ℃ = ℉
E) The closest approximation to absolute zero in ℃
F) The largest Taxicab number
G) The number that has been banished from this list
H) The largest perfect number
I) The closest approximations to the golden ratio Ï†, Euler's constant â„¯, and Ï€
J) The largest triangular number
K) The largest tetrahedral number
L) The largest Fibonacci number
M) A dozen dozen dozen (a dozen gross or great gross)
N) The largest highly composite number
O) The smallest positive nonzero quantity
P) The largest central number in Pascal's triangle
Q) The largest repdigits in bases 3 through 16 (G is the largest in base 6, and the largest in base 9 is the same as the largest in base 16)
R) The answer to life, the universe, and everything

Many require the trick of removing spaces and/or prepending a negative to the first number.

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3. History repeats itself.

4. Correction to Q: there is a larger base 6 repdigit that is not G.

5. Correction: I've found eight more primes above 100; now totaling 19.

6. PlannedChaos, do you mean(5+4)*(3*2*1) for R? That is the same as(3*2*1)*(5+4).

7. Ron, I also enjoy Futility Closet.

David, no I do not. Perhaps you think 5+4 has a different sum.

8. I was thinking of the question.

9. There is no such thing as the largest prime number. The proof is relatively simple and appears in many algebra textbooks.

I originally obtained 35 numbers, then upon seeing that it was possible to obtain 39 numbers from the first 5 positive integers, I managed to get that far as well.

I noticed that some operations appeared in clusters. That is, numbers adjacent or in the vicinity of each other differed by only one sign. Or maybe not. There are patterns, and there are deviations. Just an observation.

Never underestimate the power of parentheses! It's no accident that parentheses appear first in the PEMDAS (Please excuse my dear Aunt Sally) mnemonic for order of operations.

Likewise, there are many numbers that can be obtained more than one way. I'm sure many alternatives will appear in WS's inbox.

Lastly, while we can't use exponents or factorials (5 factorial = 5! = 5*4*3*2*1 = 120), if we could, the possibilities would be almost limitless. I recall that WS had a puzzle several years ago that required you to use 2 and some other number(s) with exponents, arithmetic operations, parentheses, etc. to come up with I believe were the numbers 1 through 10. Anyone remember that one? Ditto for one where you used the first 9 positive integers in various combinations to come up with various results. Anyone remember that one as well?

Enjoy that extra week!

26. Just like my age, I am stuck at 39.

27. On some IQ tests, one of the "tricks" to finding the answer is to apply Occam's Razor to vague or deliberately poorly-worded questions. This may be the case here. I'm still holding at 39 of 40 using the simpler of the two suggested interpretations regarding whether to use combined digits, like 54, 32, etc. or not.

28. Some statistics on the data set I generated:

With the strictest rules, 3,561 (arguably unique) equations yield 292 unique numbers. Of these, 99 are integers. Of the non-zero integers, 76 are positive and 22 are negative. Of the non-integers, 152 are positive and 41 are negative. There are 19 primes in the set. The number of ways to form each value between 1 and 40, sorted from least to greatest, is as follows:

{0, 2, 2, 4, 5, 5, 6, 6, 6, 6, 7, 10, 10, 10, 11, 17, 17, 18, 19, 19, 21, 24, 26, 32, 36, 39, 46, 51, 58, 65, 68, 73, 73, 75, 77, 78, 78, 84, 86, 96}.

With looser rules (inclusion of moving spaces and negative leading numbers), 10,096 (arguably unique) equations yield 1233 unique numbers. Of these, 515 are integers. Of the non-zero integers, half are positive and half are negative. Of the non-integers, half are positive and half are negative. There are 42 primes in the set. The number of ways to form each value between 1 and 40, sorted from least to greatest, is as follows:

{8, 10, 10, 11, 14, 15, 16, 16, 17, 19, 20, 21, 22, 24, 26, 26, 27, 28, 28, 28, 31, 31, 36, 41, 48, 54, 60, 60, 69, 74, 76, 96, 96, 98, 105, 107, 108, 132, 134, 150}.

Because parentheses are involved, the number of equations I used could probably be reduced further, but I erred on the high side so as to not potentially exclude a unique output.

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2. Another statistic: when using the strict rules, ten of the thirty-nine numbers require the use of parentheses, while the other twenty-nine do not. Under the relaxed rules, solutions without parentheses can be found for all forty.

3. John Campanelli and Jack Benny - Defying Mother Nature and Father Time.

4. GB:
I was thinking of posting Jack Benny yesterday, but thought it might be too much of a hint.

5. Hint to what? Just happens to be the age Mr. Campanelli says he's stuck at. Of course, there seems to be a lot of discussion of folks stuck Shortz of all forty. Or, should we throw out the two examples the Shortzter provided?

29. Where's Enya_and_Weird_Al_fan? My memory is that he likes this kind of puzzle.

30. I submitted an entry with all 40 numbers included. I am certain that all
but one are consistent with the rules outlined by the PM. The other one stretches the rules just a bit.

31. I got all but one with no tricks and only used / once. I think the last one requires a 'trick'.

1. Hmmmm, Charles, I used / 13 times. . .

And I agree about the trick.

2. I divided to conquer 5 times, for what that's worth. Obviously many of the numbers can be attained in more than one way.

There seems to be a consensus that all but one are manageable without tricks.

WW: you are very noble using pen and paper, I did a mix of that and a simple Excel Spreadsheet, which for me has the advantage of fewer math mistakes. Also I could type various formulas in at almost random and see what the answer is. But no massive statistical analysis per PC.

3. The more that I think about it looking at the wording of the puzzle, I don't think that last answer really is a 'trick', so I revise my answer to 40.

4. I had used / twice, but reworked one to eliminate it. As for the interpretation of the clue instruction, there is just no way Will intended to allow combined digits. Otherwise he would have done so in one of his examples. So his examples must be considered part of the instructions. And as noted by others, it just doesn't make sense to offer a puzzle with multiple ways of achieving all 40 numbers.

5. Only one of the 40 numbers requires the use of division.

6. Agreed. One-derful observation, PlannedChaos.

32. Given that it is a two-week challenge, and that Will will need to announce it again on the upcoming show, it will be interesting to see if in the second announcement of the puzzle he clears up the ambiguities discussed here. ---Rob

1. With no clear way to choose an on-air contestant and no new challenge to offer, if there is a next week's show it should be interesting.

2. I would bet they grabbed two from last week's pool and then recorded two shows. That gives Will time to travel. All I know for sure is they continue to forget to call me.

3. Me, too, Bryan. Or maybe there's a celebrity guesser?! Hillary, DT, Jill or Gary? (Better not be a geography puzzle for the latter.)

33. So is everyone agreeing that we need to show our calculations in the answer form or are we just listing the number we cannot get?

1. If you've got it, flaunt it.

2. I think all we need to state is: how many solutions there are and which number(s) don't work.
I did this just using pencil and paper. At first I vowed not to work this puzzle but it was a slow Sunday in the NFL and I decided to give this a try. Most of these came pretty easy for me. Some, I just had to walk away from and come back to later. I can't believe I stuck with it!

34. Thank you, Blainesvillains, for this helpful discussion about all the stuff I was wondering about. I, too, got all-but-one and then thought about combining digits. As noted earlier, combining digits would kind of ruin the puzzle, since you could send in a blank answer and be right. Now I’m more confident about my solution, so thanks.

I do wish Will had said the numerals 5,4,3,2, and 1 rather than the digits 5,4,3,2, and 1. Digits are supposed to get combined into numerals, so that choice of word seems like an invitation to me.

Two other thoughts. Will didn’t specifically say that you had to use all the digits. However, it’s clear that dropping a digit would be an unacceptable "trick," because you can get all 40 if you’re allowed to drop a digit.

For fun, I thought about the largest numbers possible with the intended rules—single digits with the only symbols being the 4 basic arithmetic operators plus parentheses. I believe 180 is the largest. Going down, I get 140, 125, 121, 120, 119, 115. I wonder if others believe otherwise.

1. Aha, another statistic I failed to post earlier, thanks for reminding me. The highest number that follows the strict ruleset is indeed 180, and you're also correct about the 140 to 115. The smallest is -31. Otherwise the largest is 54321 and the smallest is -54321—kind of boring, but there you have it.

2. Curious how you get -31?

5 x (4 - 3 x (2 + 1)) = -25 is the best I can do, but the computer in my brain is not very powerful.

3. What about 5 - (4*3*(2+1)) --Margaret G.

4. Ah, right you are. That's what happens when my computer hardware is hardworn.

5. Isn't the smallest -180 = (-5)*4*3*(2+1) ?

6. Charles, yes, but that uses the "cheat" that we can append a negative sign to the five. I was talking about under the most strict rules.

35. Both answers make a rather sweet play on words in my opinion ��

36. Hi Blaine,
Perhaps as a filler for week two we can do Halloween puzzles with a healthy dose of your family costumes .
We appreciate your faithfulness,
zeke creek:)

37. Longing for puzzles? I just uploaded six fresh conundrums over on Puzzleria!
A Ripping/riffing Off Shortz Slice relating to rocketry.
A campaign promise Hors d-Oeuvre;
An NFL Morsel;
A change-of-career Appetizer;
A chipmunky Slice, and finally
A timely blurby Dessert.

Go to Blaine's PUZZLE LINKS and click on "Joseph Young's Puzzleria!"

LegoFunkyChipmunky

38. I wonder if Will is going to re-run last week's puzzle? Make America grape again!

Sorry for the purple prose...

How about some peanut butter with that grape jelly?

39. This puzzle calls to mind a recent trip to the Badlands and Blackhills, unfortunately ran out of time before getting into North Dakota.

1. Lucky you, Garry Rust (jk, North Dakotans).

2. You're just fanning the flames, WW.

3. You're just fanning the flames, WW.

4. I do not repeat myself.
Why do you keep saying that?
Crooked media!

1. The book(ing) was better than the movie.

2. Still, think of the expenditure.

3. Still, think of the expenditure.

4. Yet, Blogger, fix thyself!

40. My ballot came in the mail this afternoon. I opened it right away, but have not filled it out yet as it contains 39 separate things to vote on, and I am not sure about a few of them yet and need to do some more research. I was going to mark my vote for Trump though, but I looked and looked and could not find the dictator category, just the president box.

1. Based on what you know about him in history books, what do you think Abraham Lincoln would be doing if he were alive today? (1) Writing his memoirs of the Civil War. (2) Advising the President. (3) Desperately clawing at the inside of his coffin.
- David Letterman

2. Everybody's question has to be:

What's right for my support structure?

Rebuttals?

3. SDB - More to your original point, when I turn in my advanced ballot, I hope that I won't have to put up with people who don't belong there and might be trying to 'keep order'.

4. SDB - More to your original point, when I turn in my advanced ballot, I hope that I won't have to put up with people who don't belong there and might be trying to 'keep order'.

5. My state, WA, now has total mail in or drop off ballots. I miss going to the poling place, but it is easier now. I never did see any strange stuff going on in a poling place anyway. I sure hope the nasty woman wins.

6. I agree, Trump just makes this a no-brainer!!

7. Right, but I don't think most people realize that the other Republican candidates who washed out are just as dangerous as Trump. It is so sad Bernie Sanders was sidelined, as always happens. The leaders of this country seem to be on a race to see how soon they can ruin the country beyond repair.

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42. Usually a 2 week challenge involves some sort of creativity, but many had it figured Sunday morning, and our computer experts have ostensibly verified the answer. I'd like to propose a truly creative challenge to keep us occupied and not googling, er, duckduckgoing Donald Trump 5 times a day.

The Al Smith dinner had some jokes that were truly unfunny, the creative challenge is to write alternative jokes for either Clinton or Trump. You can read the actual transcript here.

DT: [Cardinal Dolan and I have some things in common. For instance, we both run impressive properties on Fifth Avenue.] One reflects the grandeur of a magnificent building tradition of one of the greatest institutions in the history of the world.

Here's my card in case the church would like an upgrade.

DT: Matthew, Mark, Luke, and John: now there were four really great writers. They wrote some beautiful and wonderful things about their prophet. They'd never get a job at the New York Times!

DT: The Apostle Paul was another great writer, he had, what, 13 epistles? I wonder how many other epistles got deleted?

DT: You know, I sometimes think about that inevitable day when I depart and meet St. Peter at the Pearly Gates. And I know that decision is going to be rigged, and I might have to challenge it. But I am here to state that I will totally accept the results of this great and historic decision, if I get in!

HRC: I think Donald was misunderstood when he famously said he could grab women by the p---y, it was just his way of reaching out to the electorate.

43. DT - On my first day in office, June 20th, I will pass a law making Alec Baldwin and anybody else who makes fun of me, to help me build a wall around our southern neighbor, New Mexico. No matter how much the New Mexicans complain, they will pay for every red cent of it!

44. This morning, Will changed the wording of the puzzle not at all; if there are a lot of people who have already sent in the double digit answer, it has not registered in changing the puzzle for the next week.
---Rob

1. Will's still in Slovakia for the World Puzzle Championship (10/16-23). The audio quality was such that I suspect today's segment was recorded before he left last week.

45. I find it interesting that @Rob above says, "This morning, Will changed the wording of the puzzle not at all;" because evidently the editors at least at my local station edited the game as played on air. On WNYC-AM, New York, when the puzzle segment ran at 8:40 AM, the first answer given, after Will's exemplary answer, was for first names of American Presidents, and as I heard it and (nerd alert!) wrote it down, the response was George, Ulysses, Alexander (huh?), Ronald, Dwight. When the segment was re-run at 10:40 AM, the Presidents category had been excised and replaced with European countries: Germany, Ukraine, Austria, Russia, Denmark. Was this done as a courtesy to a guest from the mainstream media? Is there a conspiracy afoot? :>))

1. Oooh, NPR conspiracy. I missed the first broadcast at 6:40 a.m. here, so heard only the country list later in the morning. Why anyone would be opposed to the list of Presidents, including Alexander Hamilton, is a mystery. It doesn't seem controversial.

They were hoping for Donald for the "D?"

2. It seems that someone at WNYC knows the Presidents better than Word Woman, who may have been kidding.

WS probably would not have cleared up puzz problems even if he had been in the country.
Following a sneaking hunch, I tried out the rule of computational order (parentheses) on several people. Only one had an idea what it meant. I think that may be widespread.

3. I guess my tongue in cheek didn't come through this time. . .

4. The just-posted audio of the NPR puzzle for today still includes the Presidents category with Alexander. And Andrew was there all along. . .

And speaking of the GUARD wedding category and Us(s)her, happy earth's birthday today (officially at 9 a.m. ;-))!

5. What about Alexander Haig, in his mind, at least, after Reagan was shot: "Constitutionally, gentlemen, you have the President, the Vice President, and the Secretary of State in that order, and should the President decide he wants to transfer the helm to the Vice President, he will do so. He has not done that. As of now, I am in control here, in the White House, pending return of the Vice President and in close touch with him." Of course, he'd either overlooked the Speaker of the House and the President Pro Tem of the Senate, or planned to have John Hinckley take care of them to impress Jodie Foster.

6. Indeed.

Do you suppose the popularity of the musical "Hamilton" made Rachel, Will and guest Savannah Guthrie, believe Alexander Hamilton was President?! Will will correct that next week? We could take bets. I say yes.

7. I see the NPR Puzzles website has now printed a correction.

8. Mendo Jim:
How many replied that parentheses is a scholarly paper one writes in order to become an expert on child rearing?

46. We're approaching the 200 comment mark. Newbies, remember to click on "Load More" at the "end" of the comments to read more.

1. Yes. Or perhaps Blaine will set up a new post for week 2 of this 2-week challenge. . .

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