calling all solvers

Yesterday we were at the back to school night at my daughter’s middle school. I need to dedicate a whole post on the 2.30 hours we spent there as we retraced her steps she takes through the day, but this was something we were hit with as soon as we entered. Being the good samaritan that I am and not wanting to rob you all of the pleasure of it, here I am sharing it with you all.

Her 1st period of study is Alegbra I Honors. This is a 9th grade course that she qualified after a couple of standardized tests taken during last year, to do in 7th grade. She came home the 1st day of school and told me:

“Oh, Ms. R is horrid. I hate her”

“Okay! why? Is she strict?” I knew she was anyway from the grapevine.

“She’s just terrible Mom, you should see the rubric she gave us and then she talks like she’s joking, but she’s really completely insulting you, so we actually don’t know if we should smile or act sad. Ugh!”

“O cmon, she’s just laying down rules, you’d be fine in a few days”

“Uh-uh, I don’t think so. She’s just mean and gets perverse pleasure out of making you squirm” she scoffed and walked off.

Last night I had to agree.

As soon as we hobbled into a windowless room full of NY Giants memorabilia [she’s a NYorker and loves football] and settled into a chair, we were asked to go back out and slap a ‘parent of Jane doe’ label on us, and pick up a few sheets on the curriculum. [she almost barked at a couple who sauntered in late and told them “you need to follow instructions!’]

Then she turned on the projector and here’s what we saw. We had all of 5 minutes to figure it out, write it down and if we got it right, our kid would get a homework pass. And then she says “o, I won’t tell you the answer, coz then you’d go tell the rest of the parents. If it’s any consolation, very few get it right!’

Sure, that helps.


The following shapes form a sequence. Determine what shape comes next in sequence. All possible answers are listed on the bottom, and an explanation must accompany your choice.


Oval, Square, Hexagon, Heptagon, Octagon, ____?____


Pentagon, Hexagon, Heptagon, Octagon, Nonagan, Decagon


 No cheating! 😈

I’ve enabled comment moderation and answers hopefully will be revealed in the comments and I shall post them tomorrow morning at 9 am EST. Good luck!

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61 replies on “calling all solvers”
  1. says: Altoid

    Its Heptagon.

    Follows the number of letters of the word of the shape. For eg Oval is 4, so the next shape is a square, Square is 6 letters, so next shape is a hexagon, Hexagon is 7 letters, so next shape is a heptagon, heptagon is 8 letters, so next shape is octagon, octagon is 7 letters, so next shape again is a heptagon :).


  2. says: kutti

    the answer is heptagon again……

    oval – 4 letters, hence followed by square

    square – 6 letters, hence hexagon

    hexagon – 7 letters, hence heptagon

    heptagon – 8 letters, hence octagon

    so, octagon – 7 letters….

    answer: heptagon!!!!

    yay to me 🙂

  3. says: some body


    i go with ‘decagon’, though i can’t for the life of me figure out how the oval is supposed to represent a square or a rectangle.

    what use is my explanation if my answer is wrong? tell me my answer is right, and i can try to explain, maybe! 😉

    – s.b.

  4. says: pratap

    I know… I know…

    The sequence makes perfect sense to me… It is actually quite simple..

    Oval, Square, Hexagon, Heptagon, Octagon…

    See… all the above names are fun things for people… For example…

    Oval — this is where India got its first Cricket world cup + sachin hit the first century. So, the mention of this name itself brings a sense of joy to a whole country

    Square: again, my fav Mr. Tyson has lots of boxing matches in Madison “Square” garden. This was too easy…

    Hexagon: this is teh shape of benzene molecule, which is a highly inflammable chemical and also a fuel.. Since america likes to play with fuel… any reference to hexagon only means american fun

    Heptagon: this means 7 sided. You see, “seven” is a palindrome and also “7” looks like half a coat hanger. There are few things that give me joy more than a palindrome which looks like a coat hanger.

    Octagon is eight sided and so is octopus. actually octopus sound very squeshy! See!

    They are all just fun words. The only thing that can go with this sequence is…

    “Pentagon” where lots of US generals play lots of war games 😀

    Elementary Watson!

  5. says: Nandita

    I go two answers. None fo which are mine.
    1) First the one I think is more probable –
    The sequence is: Oval, Square, Hexagon, Heptagon, Octagon, Heptagon
    Oval has 4 letters. So the shape with 4 vetices follows.
    Square has 6 letters. So the shape with 6 vertices follows.
    So on and so forth…
    Octagon has 7 letters. So the shape with 7 sides follows..hence Heptagon.
    [From GT (]

    2) The sequence is: Oval, Square, Hexagon, Heptagon, Octagon, Decagon
    The diff between a circle and square is 4 (in terms of sides) then further between square and hexagon is 2 then 1 and 1
    So second level of diff is 4-2=2 then 2-1=1 and then 1-1=0 so the next diff here would be -1 hence we work backwards and see that it has to be decagon
    2nd level diff 2 1 0
    1st level diff 4 2 1 1
    Sides 0 4 6 7 8
    Shape Oval Square Hex Hept Oct

    The next in the series of second level of diff is -1. So adding, the next in the series for 1st level of diff is 2.
    Adding this to 8 gives 10…hence Decagon.
    From Aushin

  6. says: rads

    I know it’s earlier than 9, but the whole of last evening and this morning WordPress has been giving me login issues 😐
    *very angry and miffed as you can imagine* ..and then they have some maintenance scheduled as well..
    so jumping the gun and posting all results.

    Altoid, KC and Nandita are right 🙂

    Kiddo – Kinda creative, yet not too funny, sorry, but nice try :-p

    Ok – So you are just a number guy, not a puzzler eh? 🙂

    Rest – this was a stumbler wasn’t it?
    Apparently one parent got it right. A little consolation for us though 🙂

  7. says: rads

    KC – lol, Ingey declare panrade oru periya vishayam, idiley prize vera kekkuda? :-p

    seri, altoid and KC will get a “prize” 😉

  8. says: metlin

    Damn! You posted the solutions before I could answer. 🙁

    Not that my answer was right.

    A bunch of us sat down and figured out two solutions – one of which involves the Zeta function and the other based on a combination of Fibonacci series and the number of diagonals based on the edges available. 😛

    Now I feel like a total and complete idiot (this, right after you called my blog intelligent, sheesh!).

  9. says: some body


    misery loves company! thanks for making me feel better (now rads, you’d better thank us both, as well as the others that failed!).

    – s.b.

  10. Anyone else came up with a mathematical solution for this or is it just me (and no, i will never give that out…it’s so juicy!).

    Really nice one…..for a few seconds, it made me think that I was dumb, well I should say for a few minutes, not seconds 😉

  11. says: Metlin

    S.B. – Totally, dude! You should have seen one of the solutions that a friend of mine came up with. Were it not for Occam’s Razor, I might have been tempted to share it here.

  12. says: rads

    lol, this was totally like the snippets of conversation I heard in mutters by a few parents as we walked out. Today a few of them mailed back, calling themselves much like you guys did, after I’d sent the answer out on the letter solution.
    My inbox is plain flooded thanks to Ms. R 😀

    Metlin – I thought I said it required lateral thinking :-p

    I love such teasers – it’s so “Odyssey of the mind” concept-related outa-the-box ones! 🙂

  13. says: some body


    i didn’t want to admit how dumb i was, but what the heck.

    – i had no clue who occam is/was, and what his razor stood for (now i do – thanks google)

    – my single assumption and ‘mathematical solution’, for what they were worth (i.e., nothing) … oval has four sides; so the two sequences are 4, 6, 8 … and 4, 7, 10 … 🙂

    – s.b.

  14. says: w h

    1. Is it a math test or a vocab test??
    2. I really don’t understand what these aptitude tests mean.. there are different ways to approach the same problem, since, they are, by nature, so subjective, and each approach produces a different answer. Its not arithmetic, you know. What is ridiculous is that some people have the galls to say that a particular line of thinking is correct and another line of thinking is incorrect. I’d say, that the people who make such question papers, don’t even have the intelligence to understand that there are multiple ways to approach the same problem, each producing different results.. such is the irony of all these IQ and aptitude tests.
    Issac Asimov once wrote a nice article on these aptitude tests.. I’ll paste the stuff for you here…

    What Is Intelligence, Anyway?

    Isaac Asimov

    What is intelligence, anyway? When I was in the army, I received the kind of aptitude test that all soldiers took and, against a normal of 100, scored 160. No one at the base had ever seen a figure like that, and for two hours they made a big fuss over me. (It didn’t mean anything. The next day I was still a buck private with KP – kitchen police – as my highest duty.)

    All my life I’ve been registering scores like that, so that I have the complacent feeling that I’m highly intelligent, and I expect other people to think so too. Actually, though, don’t such scores simply mean that I am very good at answering the type of academic questions that are considered worthy of answers by people who make up the intelligence tests – people with intellectual bents similar to mine?

    For instance, I had an auto-repair man once, who, on these intelligence tests, could not possibly have scored more than 80, by my estimate. I always took it for granted that I was far more intelligent than he was. Yet, when anything went wrong with my car I hastened to him with it, watched him anxiously as he explored its vitals, and listened to his pronouncements as though they were divine oracles – and he always fixed my car.

    Well, then, suppose my auto-repair man devised questions for an intelligence test. Or suppose a carpenter did, or a farmer, or, indeed, almost anyone but an academician. By every one of those tests, I’d prove myself a moron, and I’d be a moron, too. In a world where I could not use my academic training and my verbal talents but had to do something intricate or hard, working with my hands, I would do poorly. My intelligence, then, is not absolute but is a function of the society I live in and of the fact that a small subsection of that society has managed to foist itself on the rest as an arbiter of such matters.

    Consider my auto-repair man, again. He had a habit of telling me jokes whenever he saw me. One time he raised his head from under the automobile hood to say: “Doc, a deaf-and-mute guy went into a hardware store to ask for some nails. He put two fingers together on the counter and made hammering motions with the other hand. The clerk brought him a hammer. He shook his head and pointed to the two fingers he was hammering. The clerk brought him nails. He picked out the sizes he wanted, and left. Well, doc, the next guy who came in was a blind man. He wanted scissors. How do you suppose he asked for them?”

    Indulgently, I lifted by right hand and made scissoring motions with my first two fingers. Whereupon my auto-repair man laughed raucously and said, “Why, you dumb jerk, He used his voice and asked for them.” Then he said smugly, “I’ve been trying that on all my customers today.” “Did you catch many?” I asked. “Quite a few,” he said, “but I knew for sure I’d catch you.” “Why is that?” I asked. “Because you’re so goddamned educated, doc, I knew you couldn’t be very smart.”

    And I have an uneasy feeling he had something there.
    Well, while my thougts were completely unaffected by what Isaac wrote, and they were formed long before I even knew that Isaac wrote anything other than science fiction, long before I bought Isaac’s guide to Shakespeare, I could never have expressed myself so lucidly.

    Btw.. do visit my blog.. I’ve posted something, I’m sure, you’d, and many others, would love to comment on.. awaiting your comments…

  15. says: some body


    guess what?! i was thinking of the barometer story yesterday too, except that i had read it so long ago that i was not sure whether it was a barometer or a thermometer that was involved, and so did not dare mention it here!


    growing up in india, we had (still have?) magazines like competition success review, career and competition times, science digest, science today, yada yada yada – are there similar mags at smilar levels around here? somehow popular mechanics, discover, etc. don’t really feel the same to me.

    i ask because … the barometer story – i had read it first in one of those desi magazines (and yeah, that long ago).

    w h:

    whatever asimov says, when marilyn vos savant (‘ask marilyn’) says that she has the highest iq in the world, we had all better believe her, not him :-).

    – s.b.

  16. says: w h

    metlin.. I had read that a long time earlier.. btw.. There are a lot of stories about niels bohr.. I’ve been dying to get my hands on the book “Brighter than a thousand suns” but never managed to get it..

    some body.. lol.. thats the dogma we were talking about… 😛 😀 you do have a good sense of humor…

    rads.. sorry for spoiling your page with my exhortation..

  17. says: rads

    wh – no issues absolutely! 🙂
    I’ve been debating if I should go ahead and explain or not on your earlier comment.
    I believe you are right only depending on the circumstances in which the puzzle was given in. Just as you quoted with Asimov..

    This was meant as more of a thinking puzzle than as a purely mathematical one. That was the reason I explained the story around it.

    Ms. R is the math major.
    The kids are smart 12 year olds, not all super geniuses.
    The parents are whatever they are, but what they are not are all math people.

    With just those 3 statements alone, we can safely assume that this puzzle’s answer [under the given settings] ought to be something that does not require laborious thinking and/or fundamentals from within the realms of higher abstract thinking.

    My 2 cents on what I posted.

    o btw, you should link your blog if you want us to hop on over 🙂

    sb – I am not immediately aware of exact similar ones, but I do know they exist. Will let you know.

  18. says: w h

    well.. wordpress isn’t my favourite domain, you know, and things are pretty much different here.. First, I don’t have a wordpress account, and I don’t think I’ll create one.. having too many accounts on the internet is a menace.. all sites have different password requirements.. you don’t get the same id on different websites.. the end product, you have a set of m passwords and a set of n usernames.. m * n possibilities.. Whew!!!
    Some day the internet should start a single sign-on.. the same way companies have the single sign on stuff for everything..
    btw.. i’ll just type the address of the blog here.. won’t spend much time how to hyperlink it and stuff like that on wordpress…
    Btw.. do you use an rss reader to read blogs? Google reader is a good option if you do so.. and if you don’t use an rss reader, give it a shot..

  19. says: rads

    wh – I use google reader and have added you on.

    Totally empathize on pwd situation – at work I have 3 sets, and then a few more for home. I juggle my pwds around. Not something I should be telling out here, but then again, my pwds are tricky 😀

  20. w.h,

    The way you exhorted, I am kinda reminded of a certain Dr. Hallam in one of Issac Asimov’s books “The Gods Themselves”.

    You either solve a puzzle or you dont. Thats the bottomline.

  21. says: some body


    this is a puzzle. how do you get your thumbnail image on those comments without signing on? and if you do sign on, how come rads is not clickable?

    – s.b.

  22. says: metlin

    S.B. –

    You need to sign up for a WP account for you to have those thumbnails.

    Secondly, I think Rads has not provided a website in her profile. Providing a website will automatically make your nick click-able.

    Hope this helps.

  23. Metlin,

    I never said there is a single right answer to a puzzle (e.g., Barometer question as you put a link). I said, you either solve the puzzle or you don’t.

    Also, even if you think there is no solution to a puzzle, you should be able to shred it to such a level to say its not solvable and that means you have solved the puzzle.

  24. says: Metlin

    Shadowypilgrim –

    Your boolean statement was ironic. Fortunately for the rest of us, the human mind isn’t quite satisfied with any one solution, it is usually only satisfied when the solution is the simplest one available.

    If only there were those absolutes that you speak of, in life and in math!


  25. says: some body


    that’s much better.


    i was aware of the first point you made, but not of point #2. looks like rads addressed the issue anyway. 🙂

    – s.b.

  26. says: some body


    couple of days ago, i learned about occam’s razor thanks to your comment (though i confess i’ll need to google it again before using it somewhere); today, i picked up hanlon’s, via this piece.

    i wonder if there are (m)any more razors lying around …

    – s.b.

  27. says: w h


    > You either solve a puzzle or you > dont. Thats the bottomline.

    Took these words rather seriously, and gave it more thought.. There, apparently, is no bottomline. One puzzle just leads to another. At one point of time Physics was considered a dead science, where everything that needed to be known was known… Only a few stray problems here and there.. Max Planck, then, started working on one of those last few problems – physics was considered a bad domain to work on because everything had been ‘bottomlined’. What Max discovered was that he had hit the tip of the iceberg… Honestly, there are no solutions, at least not in the real world (In an abstract mathematical world, you may find perfect solutions, and that is because you made the assumptions, you made the rules), but in a real world, where you didn’t make the rules, you hardly find that there is a solution – you just move on from one level to another..
    I’m not an expert in physics, but at one point of time Physics used to be my favourite, and, I don’t mean to be boisterous, I was pretty good at it. We say we know what magnetic field is.. We define it as something that has to do with magnetic flux and area.. great.. no qualms.. but then the question is what is magnetic flux.. Now we might find something to define magnetic flux.. but then we’d have to define something.. dig down and down.. one realizes that we all proceed on a hunch – that there is something.. We have a hunch of what that something is, but no concrete definition..

    > Also, even if you think there > is no solution to a puzzle,
    > you should be able to shred it > to such a level to say its not > solvable and that means you
    > have solved the puzzle.

    Again, you may never know, in the real world. At one point of time flight wasn’t considered possible. At another point of time space travel wasn’t considered possible. Again at another point of time time travel wasn’t considered possible. Today, it is considered extremely difficult to do.

    Now, I know that the problem given here was mathematical in nature, but then, as metlin said, the human mind isn’t quite satisfied to let that fish stay quite in the water, where it was born – no, the human mind picks it out of the water, and puts it into, say, alcohol, and see how it behaves – that is one of the hallmarks of all science (probably even mathematics) – to put something into abnormal conditions, and observe how it behaves. Now, our fish, this problem, comes out of its hypothetical mathematical world, into a real world, where there, probably are multiple solutions, where the problem is both solvable and unsolvable (Quantum states with fixed probability : the probability is fixed, but then one can never say whether it is finally solvable or not), where there are many solutions, and many non-solutions….

    Forgive me.. I tend to stray a little too much when I think…

  28. Metlin,

    Actually, my boolean statement is correct. You are interpreting things which I haven’t said.

    Your assumption for what I said is:
    1. Human mind is satisfied as soon as you solve a puzzle.
    2. Life and math is absolute.

    Infact, I never said this!

    When you solve a puzzle, you solved it, but some our minds start working on some new puzzles.

    1. What’s the easiest way to accomplish it?
    2. Is there other ways to do it?
    3. Is this the most efficient way to do it?

    (Maybe some more!)

    I also think life is absolute in a way, because ultimately, it boils down to the choice you make for any situation.

    Math is absolute, unless you are dealing with statistical quantities.

  29. says: some body

    metlin/pilgrim/w h:

    now i am really puzzled! an emphatic boolean yes. until one of you einsteins proves it that i ain’t! 😉

    – s.b.

  30. says: some body

    oops, maybe i should’ve said “truly puzzled! an emphatic boolean tru(e)ly”. oh well, that just proves how puzzled i really am!

    – s.b.

  31. says: rads

    What is happening here?! Thankfully none of it is aimed at me, and I can go back to my simpler existence, and pretend that I didn’t see any of this! 🙂

  32. says: metlin

    > Math is absolute, unless you are dealing with statistical quantities.

    Would you be kind enough to give me a generalized form for predicting the nth term of PI and giving me the proof for the Riemann hypothesis, please?


  33. Metlin,

    There is difference between something being absolute and something being an approximation. Think for a second the reason to predict the nth term of PI!

    Proof for the Riemann Hypothesis … hmm I think I will try that. Earn a Nobel prize and then reply on this blog.

    Till then, I will resolve these small puzzles Rads puts on the blog.

  34. says: w h

    > Puzzles:
    > 1. What’s the easiest way to accomplish it?
    > 2. Is there other ways to do it?
    > 3. Is this the most efficient way to do it?

    What if the ‘it’ that you see is different.. The ‘it’ happens to be hazy, and it often is, as the human mind can fuzzy-ize the ‘it’. ‘Perspectives’…
    And at times, you may not want to do it the most easiest way and the most efficient way.. (Thats why there is a word called research.. If all research was aimed at doing things the most efficient way and the easiest way, a lot of things would have never been)…

    A puzzle.. is, I’d say, something that gives you an opportunity to exercise your gray cells.. Its meant to ‘puzzle’ you, and you exercise your gray cells to ‘unpuzzle’ yourself..

    After all, even most of mathematics, is, but a perspective.. A proof is a proof because someone finds it convincing… and how much convincing is convincing.. that is perspective…
    Again, there are other categories in mathematics, where we proceed mostly by hunch… Axioms is the term and it means
    1. a self-evident truth that requires no proof.
    2. a universally accepted principle or rule.
    self-evident? whoa!!! you haven’t proved it.. how do you know its self evident.. isn’t the fact that you believe that its self evident a hunch of yours?

    To think about it, most of our Euclidean Geometry is based on these axioms.

    The only criteria we use to judge whether a proof is an actual proof or a fallacy is whether the proof is ‘sufficiently convincing’. However, inspite of all its definitions and accuracy and settings the premises and assumptions clearly before making a theory, mathematics has never clearly defined what ‘sufficiently confincing’ is.

  35. says: w h

    What I’m trying to say is, that even a proof depends on how you look at it.. Similarly a puzzle is how you look at it..

  36. says: some body


    you’ve scored a half century in the commentary section; well, actually you have only hit two singles, the rest of the contribution comes from the wide balls! 😉

    *now i need to tunnel through, else i’ll get hit by beamers*

    – s.b.

  37. says: w h

    The bad news is that nother really really wide ball is coming up.. the good news is its funny…

    It was class XI and we had this math exam in which none of us ever studied, none of us ever attended classes (first taste of college, you know, and we were experimenting with the freedom of not attending classes as long as the attendance was being maintained by proxys)..

    The question was:
    Prove that log 2 lies between 1/3 and 1/4.

    My friend proves it:
    Proof by contradiction:
    Assume opposite is true. Log 2 doesn’t lie between 1/3 and 1/4. The question asked in this question should have been “Prove that log 2 doesn’t lie between 1/3 and 1/4”. This is clearly untrue.
    Q. E. D.

  38. says: w h

    Unfortunately, the professor didn’t find the proof ‘sufficiently convincing’… Man, was he biased.. This is the most ‘sufficiently convincing’ proof you’ll eve get.. Evan a class II kid will tell you that…

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