I'm a little annoyed that the example doesn't follow the order of operations and instead must be performed left to right. It would have to be (1 + 23) ÷ 4 x 5 x 67 - 8 + 9 to work correctly in most calculators.Q: This week's challenge is a numerical one. Take the nine digits -- 1, 2, 3, 4, 5, 6, 7, 8, 9. You can group some of them and add arithmetic operations to get 2011 like this: 1 + 23 ÷ 4 x 5 x 67 - 8 + 9. If you do these operations in order from left to right, you get 2011. Well, 2011 was 15 years ago. Can you group some of the digits and add arithmetic symbols in a different way to make 2026? The digits from 1 to 9 need to stay in that order. Will knows of two different solutions, but you need to find only one of them.
I have one expression that works either way, three expressions that work left to right and one that only works using the standard order of operations. I have a feeling that Ed Pegg Jr. provided the two that work using the standard order of operations.
My only hint is 390,625.
Edit: To "brute force" this, you could take the digits 1 to 9. Then you have 5 choices of symbol (nothing, or one of the basic operators) to put in the 8 spaces between each digit. That results in 5^8 or 390,625 strings that would need to be evaluated. I submitted the first answer below because it has all the multiplication first so going left to right is the same as following the standard order of operations.
Note: The two answers following PEMDAS order are the ones that Ed Pegg Jr. provided in his Wolfram Community post which makes me think the PEMDAS answers are what were originally given to Will. I've contacted Will to see if that's the case and if Will introduced the non-standard "left to right" example and wording. I'll let you know if I hear back.
A: 12 x 34 x 5 - 6 - 7 + 8 - 9 = 2026 | PEMDAS or L to R
(1 + 2) x 3 x 4 x 56 - 7 + 8 + 9 = 2026 | L to R
(1 x 2 + 34) x 56 - 7 + 8 + 9 = 2026 | L to R
(12 - 3) x 4 x 56 - 7 + 8 + 9 = 2026 | L to R
1 + 2 + 345 x 6 - 7 x 8 + 9 = 2026 | PEMDAS
I can't imagine how to clue something like this.
ReplyDeleteNor can I... so I guess we'll just post answers on Thursday?
DeleteIf the numbers were reversed:
Delete9+8*7+654*3-2+1 = 2026 | PEMDAS
9*8-7+654*3-2+1 = 2026 | PEMDAS
9*8*7+6-5*4+3+2+1 = 2026 | Left-to-Right
98+76-5*4*3-2*1 = 2026 | Left-to-Right
98+76-5*4*3-2/1 = 2026 | Left-to-Right
To hell with math puzzles. Talk with you all next week. I'm not going to spend any more time on this one! No clues, just the truth!
ReplyDeleteMaybe you're just standing up for PEMDAS and refusing to ignore it?
DeleteSam Cooke- Wonderful world " Don't know much about trigonometry"
DeletePlease excuse my dear pseudonymic Clark
DeleteI posted the same annoyance at the end of last week's thread, Blaine.
ReplyDeleteAlso I was going to give one of the clues/remarks you gave as my clue, but I was afraid it was TMI!
I'll post my answer (which I'm almost sure is one of Blaine's) on Thursday.
I like math puzzles and wish Will would give more of them, but as I said at the end of the last thread, this type of math puzzle is like an anagram puzzle. It's not terrible to muse over it for a while, but ultimately it's pretty boring and not very creative. EPjr, I expect more.
Just 236 correct answers last week
ReplyDeleteAnother year that I'm happy to be among the 235. Names in the News scares me every time. (Maybe because I avoid the news more and more...)
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ReplyDeleteMath has rules in the order of mathematical operations… the example flouts those rules..no hint here .. just frustration..
ReplyDeleteEd Pegg Jr.’s 2011 example works using “Reverse Polish Notation” (RPN) found on HP scientific and mathematical calculators. He should at least have warned us.
ReplyDeleteAnd if you reverse RPN? 😉
DeleteI don't think the left-to-right or RPN stipulation were Ed's doing. I think Will introduced it in his example, and misunderstood Ed's two PEMDAS solutions. As a result Will's changed the puzzle. Fortunately, his revised puzzle has 4 solutions so it can still be solved.
DeleteHard to clue. In my solution, the largest number is associated with a southern state.
ReplyDeleteI don't know if I'll bother to try to solve this one or not. But as far as the on-air puzzle is concerned, I didn't know about the Japanese person and I forgot you-know-who's birth name. But I could have talked on and on about the nameless raccoon, whose story cracks me up!
ReplyDelete(I'm being deliberately vague about the two answers I got wrong to be spoiler free in case anyone listens after reading this.)
I7
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DeleteThough I didn’t see it at the time, I can see now that my remarks would have been better left unsaid. I apologize to Blain and all here.
DeleteMake that Blaine :)
DeleteI'm surprised by the deletions. Without giving anything away, I'm allowed to post that I got the answer from my big brother, but not that I got the answer from Big Brother?
DeleteSo, is it true, Blaine, that you're no longer allowing posts like "ChatGPT got it right"?
DeleteHappy MVI + DX * C/L
ReplyDeleteI did it! And it works according to the order of operations! It's all x, + and -. The largest numbers are only two digits
ReplyDeleteMy solution uses those three operations too. Maybe the other one uses division, but I don't know as I haven't found it yet.
DeleteSame here.
DeleteI've solved it, but it strikes me as a disappointing puzzle. And another disappointment: While Will did mention "lots of answers" to last week’s puzzle, I’m especially sorry he did not specifically mention the "esse" solution, which thoroughly satisfied the puzzle’s criteria.
ReplyDeleteDr.K-
DeleteI came across "esse" myself while perusing a seven letter word list, looking for four letter words in two syllables, but as I explained on thursday, did not believe it qualified as being valid, but now realize it is. When I still thought it was valid I thought about emailing Will Shortz to advise him. I have done so in the past, and he has thanked me for so doing. I suspect he may not have been informed of your "esse" answer. Keep in mind it has only been 39 years now that NPR has been running their weekly puzzle bit, and they most likely have not had time to work out all the kinks yet.
Thanks, sdb. I appreciate it. I don't think anyone has ever figured out exactly what takes place between submission and selection. It's probably something we'll never know.
DeleteYes, I agree, and that is why I have given up on that mystery so I can spend more time on the Bigfoot conundrum.
DeleteI’m great with anagrams, but can’t deal with the math puzzles. Not that I’m bad at math, just that I don’t find those kinds of puzzles interesting. So I’ll give this week a bye.
ReplyDeleteI have a solution. It was ho hum getting there.
ReplyDeleteAnyone know why Polish Notation (and the later Reverse Polish Notation (RPN)) is named after his nationality rather than the logician/philosopher Jan (nod to our jan!) Łukasiewicz?
Easier to spell!
DeleteGaah. Learning to spell someone's name? It's not that hard. Almost like puzzlers saying they don't do math puzzles.
DeleteI've had parents tell me they "don't do math" so they can't help their kids with math. I'd say "Well, I don't do reading." and smile a sardonic smile.
Ł u k a s i e w i c z
Ł u k a s i e w i c z
Ł u k a s i e w i c z
1, 2, 3, 4, 5, 6, 7, 8, 9
Smiling away. . .
Hey are you leading a cheer??
DeleteGotta be the least interesting football team in history.
As opposed to the Sicilian Defense, popularized by Gioachino Greco (né Cusentino�)--who was not Greek, despite the nickname he was given, nor Sicilian (he was from Calabria, hence his other nickname "Il Calabrese").
DeleteWhat does all this prove? That I haven't got a clue about this week's puzzle--which I mean idiomatically, not literally.
Pietro Carrera first, though. 1617. He was Sicilian.
DeleteIn "Katabasis", R F Kuang claims that all Magick grad students are into chess. I don't usually read fantasy, but I'm enjoying this one. A little Harry Potter Goes To Grad School, and a lot Harry Potter Goes To Hell. Here's a line I liked: The protagonists, Alice (like the girl who went down a rabbit hole) and Peter (like the rabbit), newly arrived in Hell (via a spell, not dead), are getting a tour of a huge academic library in the First Circle (Pride). Their guide disses the various Shades they pass: "Now, that one keeps saying he went to school in Boston and expecting everyone to know what he means. Every few years the other Shades gang up on him and brick him up behind the stacks."
DeleteI stand corrected. Let me atone by sharing my mother's second favorite joke:
DeleteA recently deceased soul is being led to his place in Hell by a demon. As they walk down a long hallway he asks the demon who is behind a door they have just passed. "Oh, that's Jews who mixed meat and dairy." "And that room?" "Muslims who drank alcohol." "What about that room?" "Catholics who ate meat on Friday." "And who's in that room?" "Episcopalians who used their salad fork for their entree." [https://www.youtube.com/watch?v=g-4-gLlF0uw]
I have a solution that works whether you use PEMDAS or do the operations in order.
ReplyDeleteMe too.
DeleteAnd so does Blaine.
Moi aussi.
DeleteMe too. But I am not particularly interested in finding the other one.
DeleteSame here.
DeleteYessiree, Bob!!!
Delete12*34*5-6-7+8-9 has all of the multiplication first, and then the addition and subtraction, so it equals 2026 whether or not PEMDAS is used.
Deletenot my cup of tea, but enjoy, y'all!
ReplyDeleteI do not enjoy winter; cold weather makes me numb. Brrr!
ReplyDeleteI was on the road all day. Got home a few hours ago, and after eating, and putting some stuff away, I figured out a solution in about 15 minutes. My solution follows PEMDAS, and has two prime numbers in it.
ReplyDeleteSame here.
DeleteMine has three, one of which...well, that's TMI. As has been pointed out, though there are two answers that pay appropriate homage to Aunt Sally.
DeleteI'm ashamed to say that my answer with three primes was 1 - 2 + 34*56 + 7*89 = 2526. I was so relieved to find that answer that I neglected to notice it was the wrong year!
DeleteWell here's a thought. Were you to ask Trump which of these two in your opinion was the greater American? Is there any doubt in your mind which one he would pick?
ReplyDelete1. Ulysses S. Grant
2. Robert E. Lee
Oh, there is very little doubt about what his answer would be. He would say: "Who?"
DeleteGrant, because Grant was a Republican and Lee was a "loser." ("America's warriors never quit, never surrender. They fight, fight, fight, and they win, win, win." Speech at Arlington Cemetery 11-11-25.)
DeleteUnless they have bone spurs.
DeleteIt is starting to feel like 390,625 ...
ReplyDelete"It was my understanding that there would be no math."
ReplyDelete-----Chevy Chase portraying Gerald Ford in a "debate" sketch on SNL, 1976
pjbSays"NotThisWeek"ToSolvingTheSundayPuzzle
For you non math lovers, check this out!
ReplyDeleteA fun math trick involves picking any single digit, repeating it three times (e.g., 777), then dividing by the sum of its digits (21). The result is always 37.
Go ahead, I know you're going to try another digit ;-).
Is that not fabulous?
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DeleteMaybe we should call you Word/Number Woman.
DeleteNodd, thanks. I would be honored.
DeleteAnd I saw your explanation. I was trying not to scare off people who say they don't enjoy math. Baby steps.
Well I'll be a preacher's beer mug!! That is really cool, WW!
DeleteScarlett, what s delightful turn of phrase about the preacher's beer mug.
DeleteI was looking up a 37 tidbit to wish a friend happy birthday today and discovered that fun fact about the prime number 37. It certainly puts 37 up high on the prime number coolness factor list.
A lovely trick, indeed! And, for math lovers, (a*111)/(a*3) = 111/3 = 37.
ReplyDeleteRight, nice, Lancek!
DeleteI think it's really interesting how sometimes you can give a mathematical *explanation* of a mathematical fact. Some proofs are explanations and others aren't.
So this is the year 37?
DeleteIf you 11+11+11, that is: 111,111, and divide it by the sum of its digits,33, you will always get 3,367.
DeleteHere's another fun fact involving the new year. Take any number. Scramble the digits to form another number. Subtract the smaller number from the larger. Add 2026. Sum the digits of the result. Sum the digits of that result. Keep going until you get a single digit number. The number will be 1.
DeleteThe lack of reactions makes me think that folks thought this was so convoluted that I must be joking. No joke; if you follow the instructions (a calculator helps) the final number will always be 1.
DeleteLancek--don't be discouraged by our silence; it is a natural reaction of some of us to this sort of mysterious property of numbers. I remember that when I first read that famous anecdote about Ramanujan's conversation with Hardy about the number 1729 (it has a wikipedia page if you don't know the story) my first reaction was "I don't think I belong to the same universe that he did." My lack of that sort of mathematical imagination makes anecdotes such as this and your fun fact all the more fascinating: the less I understand, the more I am awed.
DeleteLancek, I thought your post was one-derful!
DeleteMy solution for this year ends a trend of last year.
ReplyDeleteIn 2025, a lot of kids liked to say, "6, 7." The equation 12*34*5-6-7+8-9=2026 subtracts 6 and 7, so it ends the trend.
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ReplyDeleteI've had it! I am now boycotting Venezuela and buying my illegal drugs elsewhere. Any suggestions?
ReplyDeleteWhich republic do we get our isoamyl acetate from?
Delete(It's a way too mixed and convoluted metaphor to be truly funny, but I'm easily amused.
That brings back memories of an undergraduate organic chemistry laboratory synthesis. (c1966). The lab reeked for weeks, and it took multiple washings, with bleach, to get the aroma of over-ripe bananas out of my lab coat.
DeleteOf Cabbages and Kings and Gift of the Magi. That's bananas, O. Henry!
Delete"Time flies like an arrow ..."
Delete"God, make it a dream!"
DeleteI suspect there will be 100 or fewer correct entries this week
ReplyDeleteHrmph. Not sure I understood Blaine's solution count, but Google AI came up with what it and I believe are 4 solutions ... anybody else?!
ReplyDeleteHas anyone other than me noticed how the use AI has risen by this group? I personally think it kills the fun of doing puzzles plus reduces our ability to do them with our own brains. This is not intended to offend; just to air a pet peeve of mine.
ReplyDelete12 × 34 x 5 - 6 -7 + 8 - 9
ReplyDeleteEh, not all that exciting to solve this one. But I did learn more about Jan Lukasiewicz and RPN so I'll call it a win for early Jan (jan ;-)).
12 x 34 x 5 - 6 - 7 + 8 - 9 + 2026
ReplyDeleteNumber/math puzzles can be notoriously hard to clue, and I certainly didn’t come up with one. But I am curious to learn the significance of Blaine’s “390,625” hint.
12x34x5-6-7+8-9=2026
ReplyDelete1 + ((-2 + 34 + 5 - 6) x 7 + 8) x 9=2026
1+2+(345x6)-(7x8)+9=2026
1+2+(34x56)+7x(8+9)=2026
12x34x5-6-7+8-9
ReplyDeleteI wrote, “In my solution, the largest number is associated with a southern state.” My answer (I used the left-to-right system and, having found a solution, was unmotivated to investigate further) was 1 + 2 * 3 * 4 * 56 - 7 + 8 + 9. Fifty-Six is an oddly named town in northern Arkansas.
ReplyDeleteRob, I have been to Fifty-Six, Arkansas! Although I'd say it was perhaps evenly named ;-).
DeleteHa!
DeleteSo 56 is a town; and 67 is a state ... of mind!
DeleteSeriously, try Googling "67 meaning", I dare you!
Paul, I dared. Thanks, I think. ;-)
DeleteContinuing this numerical sequence, my mom and I visited 56, not 67, when she was 89. Quite an amazing feat for her to walk through the caverns at Blanchard Springs, AR.
(12 x 34 x 5) – (6 + 7) + 8 - 9 = 2026
ReplyDelete1 x 2 + 34 x 56 - 7 + 8 + 9 = 2026
ReplyDelete> I7
In Morse code, I7 is . . - - . . .
Using dots for 1-digit numbers and dashes for 2-digit numbers, this gives the above sequence of numbers (without operators).
12*34*5-6-7+8-9=2026
ReplyDelete12×34×5-6-7+8-9.
ReplyDeleteAgain, on this week's edition of Puzzleria!, you are all in for a treat! Our good friend and master puzzle-crafter Ecoarchitect takes center-stage with his ever-popular-and-challenging "Econfusions" puzzle feature.
ReplyDeleteThis week's Econfusions, titled "Names in the News, games that amuse... and confuse!," challenges solvers to look backward at the year 2025 – to revisit some of the high, not-so-high, and downright low points of a year growing gratefully increasingly distant in our rearview mirror.
We will upload this week's Puzzleria! very soon this very afternoon!
Also on our menus:
~ A Schpuzzle of the Week titled “Doubleday? Nay! Smith!”
~ A "No ability? NoBility! Hors d’Oeuvre" titled “Songwriting royalty reaping royalties?”
~ An “On The Ball” Slice titled "Preservation, jarringly,"
~ A "Well Underdressed Man Dessert" titled "French articles of speech & apparel," and
~ 11 Riffing Off Shortz And Pegg Entrees titled “Nine-digit discount: $20.26!” (including six penned by Nodd and one from our "prolific, talented and creative puzzle-crafting friend-or-Puzzleria!").
It just gets no better than this! Drop by!
Thank you.
LegoInvitingAllBlainesvilliansToVisitOurPuzzleSolvingParadise!
For anyone that was too busy during the holidays, our annual puzzle and video is still available for solving and watching. No math needed, but maybe some chemistry?
ReplyDelete12 x 34 x 5 - 6 - 7 + 8 - 9 = 2026
ReplyDeleteNon-Blainery solutions reported here so far:
ReplyDeleteI'm not sure these follow the content rules:
(12 x 34 x 5) – (6 + 7) + 8 - 9 = 2026
1 + ((-2 + 34 + 5 - 6) x 7 + 8) x 9 = 2026
I'm not sure these follow the order rules:
1 + 2 + (345 x 6) - (7 x 8) + 9 = 2026
1 + 2 + (34 x 56) + 7 x (8 + 9) = 2026
Solutions reported here that Google AI also found:
1 x 2 + 34 x 56 - 7 + 8 + 9 = 2026
12 x 34 x 5 - 6 - 7 + 8 + 9 = 2026
Possible solutions not reported here that Google AI found:
12 - 3 x 4 x 56 - 7 + 8 + 9 = 2026
1 + 2 x 3 x 4 x 56 - 7 + 8 + 9 = 2026
Doubtless I have screwed up somewhere, my apologies.
I don't think brackets are allowed. The puzzle explicitly mentions each permitted symbol, and doesn't mention brackets.
DeleteBTW. the latter two solutions by Google AI are essentially the same, they are based on doing the first operator - or + in LtoR order i.e. *before* the following multiplications ... So, after the 123 step both lines amount to:
Delete9 x 4 x 56 - 7 + 8 + 9 = 2026
(12 x 34 x 5) – (6 + 7) + 8 - 9 is same as #1 (just swap signs on 7)
Delete1 + ((-2 + 34 + 5 - 6) x 7 + 8) x 9 is invalid for adding parentheses
1 + 2 + (345 x 6) - (7 x 8) + 9 is #5
1 + 2 + (34 x 56) + 7 x (8 + 9) is invalid for adding parentheses
1 x 2 + 34 x 56 - 7 + 8 + 9 is #3 (L to R evaluation)
12 x 34 x 5 - 6 - 7 + 8 + 9 is also #1
12 - 3 x 4 x 56 - 7 + 8 + 9 is #4 (L to R evaluation)
1 + 2 x 3 x 4 x 56 - 7 + 8 + 9 is #2 (L to R evaluation)
And just to clarify, the parentheses I added in #2, #3 and #4 are to explain how they would have to appear to force left to right evaluation overriding standard PEMDAS order. But if you drop the parentheses they evaluate L to R as 2026.
DeleteI think that anyone who found a solution to this week's puzzle by brain and not by computer is heroic! But, the quest to find ALL the solutions is a different project going beyond Will's puzzle, and for which the computer seems a natural tool.
ReplyDeleteIn its analysis, Google AI divided the 5 solutions into two different categories, one with a unique solution and the other being a set of four. It did a polished presentation of its results, I will just quote it here:
Interpretation A: The "Operator" Method
Rule: Strict Left-to-Right. The || is a functional step that "glues" the next digit onto the current running total.
The Solution:
1 + 2 + (345 x 6) - (7 x 8) + 9 = 2026 Blaine's #5
Interpretation B: The "First-Pass Glue" Method
Rule: Digits are "glued" together into blocks (like 34 or 56) before the math begins.
The Solutions:
1 x 2 + 34 x 56 - 7 + 8 + 9 = 2026 Blaine's #3
12 - 3 x 4 x 56 - 7 + 8 + 9 = 2026 Blaine's #4
1 + 2 x 3 x 4 x 56 - 7 + 8 + 9 = 2026 Blaine's #2
12 x 34 x 5 - 6 - 7 + 8 + 9 = 2026 Blaine's #1