Math Quiz One

{Vicky - this page is a little long. Can you edit it? -- Chimpa}

Here's a MathQuizOneQthree link to help people get to the crux faster -- GeraldYoung

(true true - really too long)

It will be shortened if someone solves Q3 (why not have a go at it?) or if I decide to give the answer to Q3 in 2004. It's fairly easy to see what may be relevant at present. -- vk

Rules: 1) Anonymous answers will be deleted, 2) Correct answers qualify you for the hall of fame, 3) Incorrect answers qualify you for the hall of shame! Those who get the right answer for the wrong reason(s) may qualify for the hall of shame.

Please don't give hints or explanations - it spoils things for others.

The 'current' question is Q3. [Apparently, it's less easy than it looks, but it is a fair question. Feel free to let me know if you dislike it, however! It can, of course, be answered correctly - if you're careful.] -- VickiKerr

Q3 remains unsolved since December 2000. You can skip to Q3 or perhaps benefit by reviewing the already-solved Q1 and Q2.

I solved 2 parts of Q3 ages ago, and commented that one of them is ill-defined.

You thought you solved... but I never confirmed that! So possibly you did make some genuine mathematical slip somewhere, in which case your refusal on T1 wasn't critical. I've slightly clarified T1 because of emails someone sent recently; be assured that no-one is failed due to minor interpretation problems anyway. -- vk
Regarding the next question, vote 'a' for an easy geometrical construction question, or 'b' for some algebraic axioms, with a request to 'see what you can do with these' (making it open-ended, with advantage to those who get in first).

Votes: I vote for lateral as Q3 is. However, I rejoin those who persistantly tell you that in failing to acknowledge that this page is an incomprehensible mess to all except you, you risk being placed in the category alongside my near and dear relatives with eleven drawers kept full of string. You have already said you don't need an about page etc.etc but my dear friend you need something.... -- AndrewCates

Problem-solvings skills include the ability to spot what may be relevant. -- vk
Q1: What number is the first blank in the well-known sequence below?

60, blank, blank, 120, blank, blank, 140, 144, blank, 150, ...
Q2: Why is the sequence "well-known"?
Q3: Consider the three mathematical tasks below. How many can be accomplished? (To avoid helping others, do not give any comments yet on the individual tasks, but those who submit the correct total may be asked for such details later!)

Task 1: Using only the digits 1, 2 and 3 (exactly once each) and standard mathematical symbols (but no letters), form a mathematical expression with value exactly 19. Please note that a symbol which rounds (or truncates) a non-integer value is not allowed for this task (not sufficiently standard). (I apologize for not mentioning this rule originally.) Note: generalizations of the factorial function are taught in some countries, but I doubt if they are widely known elsewhere, so they are not allowed (except during May 2002 for the one emailer who alerted me to them). (See also note at end of page.)

Task 2: The expression 9*7-4*8-2 (where * is the multiplication operator) has value 29 (since multiplication takes precedence over subtraction). Now suppose you evaluate using a calculator of the older type which implements add, subtract, multiply and divide in the order they are entered, so that the expression has value 470. Replace each multiplication and subtraction symbol with a mathematical symbol of your choice (without using letters, brackets, or digits) so that the value of the expression becomes 29 again. See MathQuizRuby.

[NicholasBasso asked if use of 10^x (which is on his calculator) is allowed. He did not say which task he had in mind. 'No letters' bars the function for both T1 and T2. I didn't ban the '%' symbol for T1 because I could see no way to use it to obtain 19. For most calculators, '%' is not available or acts as an expression terminator, so it would not be acceptable for T2. You cannot invent new symbols for T1 (because defining them would require letters, and new symbols could hardly be called standard). General exponentiation (eg., 2^3) is allowed for T1, but not for T2 (because most calculators with a key for it label the key using letters). -- VickiKerr]

Task 3: Given that v is a function (of t) satisfying the differential equation

         t  dv/dt = 5 + 4 sin(v) + 3 cos(v) (t < 10),

and that for t = 2, v is zero, prove that when t is the negative square root of 3, v is a rational multiple of pi.

Could you be more precise about the terms "standard mathematical symbols" and "a mathematical symbol of your choice"? (I can give examples of constructions that might be dubious, but I don't want to give any games away if they are part of what you intend...)

I certainly didn't mean symbols not usually seen in school textbooks. Indeed, for the second task, it would not be acceptable to use a symbol that cannot be found on most calculators. If still in doubt, email me.

By "symbols" do you mean to say "operators"? -- AndyPierce [See below.]

The only other restriction on 'symbol' is that two symbols cannot count as one. -- VickiKerr

I have a question about the operators allowed for Q3, task 1 (Vicky, I sent you an e-mail) -- RolandKaufmann ["Yes" to both your queries. -- vk]

A3: 2 of 3 -- DavidLong [His second email improved on his first, but he missed a separate error (which I won't detail yet to avoid giving clues) -- vk] [n.b.: hence further tries not allowed, David]

A3: two are possible, one is not -- DaveWhipp.

[Dave: may I point out that the negative square root of 3 is not the same as the square root of minus three? You may revise your answer in respect of T3 if you wish. -- VickiKerr]

A3: all three are possible, according to GarethMcCaughan (see below), but is he right? No! -- vk

A3: all three are possible -- MagnusJosefsson [Brave attempt. -- vk]

Solvers are requested to email a short, text-only, explanation of their answer to me. To do so, (and draw my attention by editing this page). As this is a secondary email address, it may take a little while before I react. [I will not usually reply by email. -- vk]

just mailed you at -- gp

An email from DavidLong prompted me to revise T1. He had used "ceiling". Although this uses letters, he assures me there is a standard symbol for this function. However, I had difficulty finding it in textbooks so I have barred it explicitly. -- vk

Ceiling looks like [n] except that the brackets are only the vertical bars and the topmost horizontal bars. Floor has the bottommost horizontal bars. I can do it using funky notation too, but it simplifies to BEDMAS.

Acknowledged, but it's simply a matter of whether the symbols are included in a published standard (eg., for schools), and have also been accepted by authors of textbooks (other than of books on number theory).

Ceiling and floor are also commonly used in numerical analysis and computer science, and are crucial in the notation of common algorithmic analysis.

Is Q3 too hard? Many have read it, yet almost no answers as yet.

Perhaps T3 is incongruous with T1 & T2? [True, but intentional]

I have done two of the questions (including #3) and have an unproved conjecture about the other one. #3 was the first I did, and I did not find it difficult. Of course, for all I know my solution may be wrong... -- GarethMcCaughan

[I have made the notes clearer. Hopefully, this will avoid confusion. Normally, only one attempt allowed, so take due care. Beyond that, I can't comment without giving hints. For T2, choose your mathematical symbols from those on older-style, cheap calculators (hence no fancy statistical/graphics symbols). The last sentence added for the benefit of GarethMcCaughan. -- VickiKerr]

I have now answered all three parts of Quiz Three and mailed Vicki. I am deliberately not putting my count of do-able tasks on this page, because I know that I have been loth to read it for fear of getting information I didn't want from other people's counts. (As it turns out, no one has submitted an answer that satisfied Vicki, so my worry has been unnecessary, but even so...) -- GarethMcCaughan [See note above. -- vk] [Please read your mail again. -- gjm] Your opening paragraph is within the rules. I quickly made the task rules clear enough, while trying to avoid giving hints. However, will you now take the plunge and commit yourself to an answer on this page? -- VickiKerr

I don't understand in what relevant respect I have not "taken the plunge". I am content for the two e-mails I have sent you (one before and one after your latest clarification of the rules) to be "binding". I prefer not to say how many of the tasks I think can be done, because that would (if I'm right) give other readers of this page information that they would prefer not to have.

It sounds as if you think I am trying to avoid committing myself, or trying to avoid exposing myself. I hereby undertake to send details of my answers to anyone who requests them (within, say, the next couple of weeks). I am not trying to avoid commitment, scrutiny, or risk.

My "opening paragraph" may perhaps have given a wrong impression. Its only point was to answer the question "Is Q3 too hard?" nothing more at all. -- GarethMcCaughan [I meant the opening paragraph of your email! -- vk]

I'm just sticking to the declared rules. Since you said earlier on this page that you might be wrong, I don't really have any choice. Don't worry about the possibility that your total might help others. Perhaps others have your answers in mind as well. Someone has to be first. -- VickiKerr

I still think there is some misunderstanding going on here. However: I hereby declare it to be my opinion that all three of the tasks can be achieved. I still might be wrong (so might we all, always), though it doesn't seem likely. -- GarethMcCaughan

A3: the answer is the least significant digit of 42. And no, I don't think it was too hard. -- RolandKaufmann

A3: three (by Ben Tilly - see his note later. -- vk)
Since questions 1 and 2 have been answered, I give the answers in MathQuizHistory, and comments will be transferred there or deleted.

A2: It's a function on integers, but starts at three, that is, f(3)=60. -- AndyJewell
Fame: AndyJewell (Q1,Q2), Sunir, DaveSmith (Q1,Q2), StephanHouben (TOF), EricHerman (TOF), MatthewWilbert (Q2)

TOF=Taken On Faith
Shame:AndrewMcMeikan (85 claimed), Sunir (on Q2, by own admission!), DavidLong (Q3), DaveWhipp (on Q3 (subject to note)), GarethMcCaughan, RolandKaufmann, MagnusJosefsson, MichelDauchez (on Q3 - broke or misunderstood the rules), BenTilly (on Q3 - broke a rule and made mathematical slips), J├╝rgenBlumeNienhaus (Q3), AaronHill? (on Q3), AndrewCates (on Q3), GeraldYoung (on Q3), DarrickWiebe? (on Q3)

Once you're in both the fame and shame lists, maybe the next victory or failure could remove you from one of them? Yes, probably. I've had that in mind for a while. -- vk

I hope this isn't one of those (AmericanCulturalAssumption) New York train stop sequence puzzles.

Don't worry - it's strictly mathematics. It wouldn't be fair otherwise. -- vk

Is the sequence definitely CORRECT? [Yes] The last number seems odd. -- amc

The last number is actually even. ;) Did you graph it? -- anon.

I'm still unsure whether anon realizes why the sequence is well-known -- VickiKerr.

I don't, but I'm going to guess it has something to do with harmonics and waves. It's 180 as in 180 degrees (pi radians). However, that sounds wrong because the ratios don't match the musical scale properly. On the other hand, the sequence is roughly the inverse of the harmonic sequence. -- anon

OK, I got it now, but I don't see how graphing the sequence would have been of any help. -- StephanHouben

It's a general heuristic to solving sequences. The sequence is solved if you can find a transformation that makes it linear with one of the axes (preferably the x axis) being n and the other being T(n).

Also, graphing the sequence gives you a lot of information about its form by visual-pattern recognition. In this case, it seems as if the sequence is either bell-shaped (like a sine curve) or asymptotic to some y=c. In the latter case, it seems to be some sort of flipped decay curve, so you have to find the decay rate and the means to flip it. I got the recurrence which I later solved to give the actual formula. -- anon

Did you find the geometric interpretation? -- sh

Yes. Eric's hint gave it away for me like Ron. Good work, Eric. -- anon

To appeal a result, place details on my home page. -- vk
TaralDragon seems to have given up on T3! -- vk

Unfortunately, I don't have the skills to answer T3 properly. I can limit the answer, and guess at it, but I don't know for sure. -- td

Hmmm. That depends. I can't say more than that without giving a hint. -- vk
Where are the solutions? -- Curious. In MathQuizHistory, as stated above!
Progress is somewhat slow! Surely some more attempts can be made at Q3! Is there a geometrical theme?

Vote for harder (h) or easier(e) or same(s) next time. (I don't guarantee to abide by the vote.)

Votes: ssehsssssees
For future reference, vote b to bar calculus or a to permit it.

Votes: baaaaabba
MathQuiz (originally)

JohnWethington gives this page five stars.

Yep it is a beast

I just came across this page. I worked out the first one, but then saw that this was old hat :) I had a brief look at Q3. My results are

1) not well defined. I won't play with this one without a specific list of operators that are allowed. [Can you email your suggested list? -- vk]

2) is not possible, if I have understood the algebraic grouping rules correctly (these could be made explicit, vk!!)

3) is true.

I won't place my reasoning here in case others still want to play with them. Shall I send you (vk) my results? Or do you wish to define the first part rigorously? -- AnonymousMathy?

If in doubt, email your query or partial solution to me. Nobody fails on some minor quibble. Most solvers found the notes above clear enough. If no email arrives, I'll remove your comments after a few days. -- VickiKerr

Me again. To clear up my comments above: I worked out Q1 (and Q2). What I had trouble with was the definition of Q3, part 1. I have no doubts about my solutions; were you asking for proof, or offering to check people?

To enter, give your total (0, 1, 2, or 3) of tasks that can be done, and your name, and send me an email explaining in brief how you arrived at your answer. Full proofs are not necessary, but don't expect success by guesswork alone.

Q3 part1) This seeks a 'normal' mathematical expression - no 'pseudo-maths' or symbols so specialized that they would probably not appear in a school textbook. Square brackets are equivalent to their round counterparts. -- vk

You still haven't answered my question about the allowable operators. From comments on this page I would guess {+,-,*,/, ! %,sqrt} are there any more? And one '1', one '2', one '3', no other numerals. Correct? I'll accept an answer based on that, yes. -- vk

Q3 part2) If you're trying all possibilities, you might find it easier to think about how to get an odd number rather than an even number. -- vk That is essentially what I said in my comment... I was obliquely pointing to the solution. [Oh. Perhaps too obliquely for me, but it shouldn't matter now. -- vk]

Q3 part3) Tough calculus or easy trigonometry? You decide! -- vk How about easy calculus? ;) Anyway, I probably won't look at this again for a week or so - I have to defend a thesis next week... [It might mislead to suggest the calculus is easy, as mistakes are still possible. -- vk]

Hi there, don't know what the answer is but this is fun!! :) heehee

This is a strange page, interesting but strange.

Q3 T1) (1+2)^3 in base 18 :) -- soobedaar [No "8" allowed! -- vk]

No 8 used! The expression is (1+2)^3 but the base used is 18 (as against 10) -- soobedaar

You are not allowed "8" in "forming the expression". That includes specifying a non-standard base. Anyway, you're supposed to give a count, not individual solutions. -- vk
Q1 and Q2 Have been solved (see MathQuizHistory) so that only leaves Q3. Q3 says How many tasks can be solved, so there could be four people; one says 0 tasks can be solved, one says 1 task can be solved, one says 2 tasks can be solved, and the last one says all 3 tasks can be solved. One will be right. So Q3 could easily be solved. -- KevinKostlan

Quite correct, Kevin. However, to get in the Hall of Fame, you need not just the right answer, but the right reasons (by email) why it's the right answer. -- vk
What do you mean by standard mathematical symbols in q3?
Two can be done. Use [hint removed by vk] & easy calculus. I had to learn about differentials before saying that. Mail follows. -- MichelDauchez

Hall of game: vk. I did make 2 mistakes, once giving hints (I was proud and supposed I solved all); then reading rules too quickly. I got shot. Second chance? If only I had ... time, or ideas, or what misses to us all - success.

I sent my solutions to Q3. Each had a different stupid trick to it, and I am not sure that you will accept a couple of the tricks that I used. (Update: My answer was that all three can be done.) -- BenTilly

Ben, you ignored the "no digits" stipulation for task 2, and slipped up in your trigonometry and substitution for task 3. -- VickiKerr

Are the shift operators << and >> allowed for Q3, part 1?

That's a good question. It's about time someone asked about such symbols. My answer is that <, <<, >, >>, ++, --, etc. are specialized symbols related to some programming languages or used in number theory, but are not symbols commonly found in school mathematics textbooks, so they are not allowed. -- vk

A3: All 3 are possible! J├╝rgenBlumeNienhaus

A very good try, but you made a mathematical error somewhere. -- vk

OK OK, I was blind. 2 of 3. -- jbn
This is just silly. VickiKerr should explicitly enumerate all of the allowed operations for Q3:T1 and Q3:T2. When she says that she will "accept answers" based on {+,-,*,/,!,%,sqrt} (for Q3:T1, right?), rather than volunteer the full list of acceptable operations explicity, VickiKerr is pointlessly making the problem drag on, with questions followed by one clarification after another. Having unspecified rules which are divulged a little bit at a time, upon violations, wastes everyone's time. It is irresponsible to present a problem for which you have not specified all of the rules, and you should consider how people might possibly misinterpret the rules when you are writing the specification.

If only software manuals were all well-organized, with everything beautifully clear! I know the answers, and I'm certain the quiz is fair (and quite easy to boot). As already stated near the top of the page, "no-one is failed due to minor interpretation problems", so quit grumbling - you don't need to be spoon-fed. -- vk
A3: I have answers for all three tasks, so I say the answer is 3. -- AaronHill?

Not a bad try, Aaron, but 2! and 2# both equal 2 (whereas p_2# = 6). -- vk

How do I send you answers, Vicki? I just stumbled into this page and became enthused with your problems. -- Sinatra

There's already a link on this page for sending me email. -- vk

Got a Q about Q3: Is "&" a common(allowed) symbol? -- Mike No. What would it mean? -- vk
Isn't (spoiler deleted) or am I missing something? Hall of shame here I come... -- AndrewCates

Cool. But you need to correct your notation. -- dm

Thanks. -- ac

No credit for partial solutions - please take note of the instructions (at the top) for solvers, and hence avoid spoiling things for others. -- vk

When you say at top you mean buried in a very long waffle but I suppose I should have read it. -- ac

Ok for the Hall of Shame. They are all definitely doable. Although I used a sledgehammer for the last one so I may have blundered. -- AndrewCates

You must have read the question, which makes it clear that Q3 calls for a count, with justification supplied by email. So I await your email, on receipt of which you will be considered to have submitted "3" as your answer. -- vk

Email sent - how do you get back to me? Incidentally, you are a bit of a rule-changer. Above, you only said "but those who submit the correct total may be asked for such details later"; now, apparently, I need the details up front.

That assumes you submitted the correct total! You made some mathematical slips, at least one of which was crucial. A good attempt though. -- vk

A3) The count is 3. Email is sent. GeraldYoung --- oops, 2. Late night math hiccup. -- GeraldYoung

I've stopped working on this 7-1-04. I submit a final count as 1 and email follows. -- GeraldYoung

You made at least one mathematical error. -- vk
Can I test answer here? -- GeraldYoung

I've emailed my answer for Q3: 2 -- DarrickWiebe?

This is the first MathQuiz.

Just browsing around and checking on how easy this could be. -- Yuskar

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