Wikipedia:Reference desk/Archives/Mathematics/2011 January 12

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January 12

Maths competition puzzle

Hello - today I did a maths competition and although I understood how to do the first five questions (whether I got them right is a different matter...) the sixth had me stumped. It was like this (paraphrase follows): What is the only polynomial function f(x) with integral coefficients and of degree 1 or greater such that f(√3+√2)=√3 - √2. HOw would I solve this? THanks. 24.92.70.160 (talk) 03:30, 12 January 2011 (UTC)

Are you sure that the question asked for "degree 1 or greater"? There are infinitely many such polynomials. But there is exactly one with degree 3 or less.

Set x=√3+√2 and work out the powers of x as linear combinations of √2, √3 and √2√3:

x^1 = √2 + √3

x^2 = 5 + 2√2√3

x^3 = 11√2 + 9√3

Treat these identities as equations with √2, √3 and √2√3 as unknowns and x as a constant -- solving those will allow you to express the roots (and so also their combinations) as polynomials of degree at most 3 with rational coefficients. In this case we don't even need to bother with a systematic solution: we see immediately that x³-9x = 2√2, and therefore √3-√2 = x-(x³-9x) = 10x-x³. Luckily this has integral coefficients, as requested, or we'd be in trouble!

Furthermore

x^4 = 49 + 20√2√3

so

x^4 - 10x^2 + 1 = 0

and we can add any multiple of this polynomial to 10x-x³ without affecting f(√3+√2)=√3-√2. –Henning Makholm (talk) 04:29, 12 January 2011 (UTC)

Weird length issue

Consider a collection of subintervals of [0,1] whose length add up to less than 1. Now its very clear that the union of all such intervels cannot cover [0,1], because we can inductively paste them end to end to maximise the stuff they cover. This intuition also works for rationals, but we can also cover the rationals with intervals that sum to arbitrarily small length (the typical ε/2^n argument to show the rationals have measure 0). Can someone explain what's going on? Money is tight (talk) 06:04, 12 January 2011 (UTC)

I'm not sure what there is to explain; your analysis is correct. You've discovered how to create an open dense set of arbitrarily small measure. If you take an appropriate countable intersection, you create a comeagre set of measure 0. The complement, of course, is a meagre set of measure 1. The lesson here is that "comeager" and "full measure" are orthogonal methods of defining bigness.--203.97.79.114 (talk) 06:59, 12 January 2011 (UTC)

I was looking again at your question, and I noticed "this intuition also works for rationals". Do you mean that this shows that the rationals cannot be covered by a collection of intervals of total length less than 1? That's incorrect. The formalization of your argument in the second sentence is by measure: Since measure is countably additive, the measure of the union is the sum of the measures, which is less than 1 by assumption. So the union cannot contain the unit interval, which has measure 1. The rationals, on the other hand, have measure 0, so there is no contradiction with them being covered by a union of small measure.--203.97.79.114 (talk) 07:15, 12 January 2011 (UTC)

Nice little paradox. I think its resolution is that the intuitive argument does not actually work for the reals (even though it happens to reach a true conclusion in that case (except what if there are uncountably many intervals?)). Specifically, who says that pasting the intervals "end to end" will maximize the covered length? If we can have infinitely many intervals, it is certainly possible to cover [0,1] with closed intervals without any overlapping interiors such that many "right end" numbers are not also "left end" ones. –Henning Makholm (talk) 11:43, 12 January 2011 (UTC)

1 2 3 4 5 6 7 8 9 = 100

I came across a variation of the old puzzle where you're supposed to insert operation signs in the gaps between the digits. The variation is:

Write 100 with all the digits 1-9, but with only one symbol denoting an operation. The only hint given is that fractional form is required.

I've been trying this for a long time and I'm still stuck. 220.255.1.75 (talk) 12:49, 12 January 2011 (UTC)

What is the name for such puzzles? 220.255.1.56 (talk) 14:03, 12 January 2011 (UTC)

I'm not sure if this is possible. I think not. 220.255.1.33 (talk) 14:13, 12 January 2011 (UTC)

I think it requires the use of decimal points. 220.255.1.28 (talk) 14:17, 12 January 2011 (UTC)

I can't tell you the name, but I found that 8 x 9 = 72; 72 + 7 = 79; 79 + 6 = 85; 85 + 5 = 90; 90 + 4 = 94; 94 + 3 = 97; 97 + 2 = 99; 99 + 1 = 100. Is that what you are looking for or did I misunderstand your question? Toshio Yamaguchi (talk) 14:31, 12 January 2011 (UTC)

This is a common computer programming problem for exams or contests. It requires nothing more than a for loop to test all combinations and locate: 1/2+3*4*5+6+7+8+9=100. -- kainaw™ 14:34, 12 January 2011 (UTC)

(account created) I meant with just one operation symbol, actually...I'm not sure whether a decimal point counts as an "operation symbol". Lanthanum-138 (talk) 14:37, 12 January 2011 (UTC)

By "one operation symbol", I mean that within all the spaces, there's only one symbol denoting an operation. I'm not sure about whether decimal points are allowed, but they probably are! Lanthanum-138 (talk) 14:39, 12 January 2011 (UTC)

Using only 1 operation from +, -, *, /, there are actually many solutions:

1/2+3*4*5+6+7+8+9=100                        
1+2+3+4+5*6-7+8+9=100                        
1*2*3+4+5*6-7+8+9=100                        
1*2+3*4-5*6-7+8+9=100                        
1*2-3+4*5*6-7+8+9=100                        
1*2+3*4*5+6-7-8+9=100

Those are just a few. -- kainaw™ 14:41, 12 January 2011 (UTC)

The problem actually asked for giving just one operation symbol (not one repeated many times), with concentation allowed, and with decimal points allowed, now I've checked it again. Lanthanum-138 (talk) 14:47, 12 January 2011 (UTC)

So you mean you have any two combinations of the digits 0-9, like 10243 and 59876 and you now want to connect these two numbers by an operation symbol, such that for example 51423 ? 60978 = 100? Toshio Yamaguchi (talk) 15:38, 12 January 2011 (UTC)

87.659 + 12.34 = 100 with a little measurement error.—Emil J. 18:30, 12 January 2011 (UTC)

I've managed 101 with "123.4-5.6-7.8-9". (Kainaw you've got lots of different symbols, you are using * and + and - in your last line.) -- SGBailey (talk) 16:08, 12 January 2011 (UTC)

I am only using "one symbol denoting an operation" for each operation. You are using "one symbol denoting an operation" for all operations. In both cases, we do not have "an operation". We have multiple operations. If you want to nit-pick, you used three subtraction operations, not "an operation". That is the confusion of the question. -- kainaw™ 17:54, 12 January 2011 (UTC)

It is unclear to me what you take the restriction to mean. Does it exclude anything at all? Could you provide an example of a meaningful completion that would not meet your understanding of "one symbol denoting an operation"? –Henning Makholm (talk) 18:01, 12 January 2011 (UTC)

It could mean that two-symbol operations (like >>) are not allowed. The main issue is how many operations in total are allowed. One operation and that is all? One type of operation as many times as you like? I read it as one symbol per operation - which is a very common usage when this appears in exams and contests in computer programming. -- kainaw™ 18:30, 12 January 2011 (UTC)

I interpret "Write 100 with all the digits 1-9, but with only one symbol denoting an operation" to be a pretty clear-cut global constraint—the things that you are allowed to write are the digits 1 through 9 and a single operation symbol, once in the whole expression. The hint that "fractional form is required" seems to imply that decimal points are also allowed. Also note that nowhere in the problem does it say the digits must come in order. It seems that the best suggestion we have so far is Emil's "87.659 + 12.34 = 100 with a little measurement error." —Bkell (talk) 22:23, 12 January 2011 (UTC)

I wonder, how many of these operation symbols are there in ${\frac {\log 4951238}{\log {\frac {7}{6}}}}$ ? How many if you also add a floor because you need more precision? 78.92.11.106 (talk) 20:59, 12 January 2011 (UTC)

I think the puzzle meant that an operation refers to one of +, -, * or /. So, there would be 1 2 operations either way. Lanthanum-138 (talk) 07:10, 13 January 2011 (UTC)

Or how about removing that measurement error with an extra dot as in $\textstyle 87.65{\dot {9}}+12.34$ ? Dmcq (talk) 23:55, 12 January 2011 (UTC)

I suppose the hidden requirement is that all digits (1 trough 9) be written in the order given – and I can't see so far what the solution could be. However IF the puzzle actually allows to reorder digits, then the answer could be e.g. $2_{134567}+98=100$ CiaPan (talk) 01:42, 13 January 2011 (UTC)

Avoiding 9 recurring the same trick can be used for $98.76{\dot {5}}+1.23{\dot {4}}$ Dmcq (talk) 02:41, 13 January 2011 (UTC)

OK, based on the context of the puzzle dots for recurring decimals is probably allowed. But what exactly is an "operation symbol" anyway? (Generally, I mean.) Lanthanum-138 (talk) 07:13, 13 January 2011 (UTC)

cylindrical/spherical integration

Unfortunately this type seems to always be lacking as an example in maths textbooks and its also been awhile since i did integration.

I have an equation f(r,z,θ) (ie in cylindrical coords)that i wish to integrate over some volume, but i only know my limits in spherical corods (R,φ,θ)

Do i have to convert f() to spherical and integrate that? or can i convert my limits from spherical to cylindrical? If i change the limits i get r=Rsin(φ) and z=Rcos(φ) which i cant see how i would use. Thanks --81.147.36.98 (talk) 22:59, 12 January 2011 (UTC)

First, note (as you probably already have) that $\theta$ comes along for free. You can substitute $f(r,z,\theta )=f(r(R,\phi ),z(R,\phi ),\theta )$ (and then still integrate $f(\dots )r^{2}\sin \phi \,dr\,d\phi \,d\theta$ in the spherical style), which I think is your first suggestion. If you change the limits, then even in the simplest case where it started as $\int _{\theta _{1}}^{\theta _{2}}\int _{\phi _{1}}^{\phi _{2}}\int _{R_{1}}^{R_{2}}f(\dots )R^{2}\sin \phi \,dR\,d\phi \,d\theta$ you'll have to consider several different cases of lines of constant r and θ that may even intersect the spherical "plate" in two different segments. But you can eventually get it to $\int _{\theta _{1}}^{\theta _{2}}\int _{r_{\min }}^{r_{\max }}\sum _{k}\int _{z_{k_{-}}(r)}^{z_{k_{+}}(r)}f(r,z,\theta )r\,dz\,dr\,d\theta$, where $r_{\min }=R_{1}\min _{\phi }\sin \phi \,$, $r_{\max }=R_{2}\max _{\phi }\sin \phi \,$, each segment $(r,z_{k_{-}}(r),\theta )$–$(r,z_{k_{+}}(r),\theta )$ intersects the boundary of your region at both ends, and for some (small) values of r you may have two values of k while you have only one for others. --Tardis (talk) 00:00, 13 January 2011 (UTC)