Burning the Candle

[Roark]

I Love Math!

My daughter, age 9, has a book of Mensa puzzles for kids. She hit me with this one:

A guy recycles candle stubs and can make 1 new candle out of 7 stubs. How many candles can he make out of 679 stubs? The solution is simple:

Stubs, Candles Made, Leftover Stubs
679, 97, 0
97, 13, 6
19, 2, 5
7, 1, 0
Answer: 113

So, I’m teaching myself calculus and I know that I can find a general solution for this problem. Well…not quite. I know that the answer is a series:

f(x) = { (x/(7^1)) + (x/(7^2)) + (x/(7^3)) + ... + (x/(7^n)) }

Testing the series with some actual values I get:

x, f(x)
679, 113.120
15,139, 2,552.103
3,020,302, 503,174.011

(The fractional portion is the stub left over after burning the last candle)

What’s the general solution?

[dave]

Originally posted by Roark
A guy recycles candle stubs and can make 1 new candle out of 7 stubs. How many candles can he make out of 679 stubs?

From the question, surely the answer is 679/7 or am I missing something?

[Roark]

Your missing something. 679 / 7 = 97 candles that you can then burn to make 97 stubs, those 97 stubs make 13 more candles with 6 stubs left over...and so on.

[dave]

Originally posted by Roark
Your missing something. 679 / 7 = 97 candles that you can then burn to make 97 stubs, those 97 stubs make 13 more candles with 6 stubs left over...and so on.

This is why I hate mensa questions. Can't think of a way to sum that series offhand, I'll have a look at it though.

[dave]

In fact, it's dead easy.

f(x) = x*sum(r=1->x) 7^(-r)
= x * ( sum(r=1->inf) 7^(-r) - sum(r=n+1->inf) 7^(-r)

Basically, I've split the sum up into 2 seperate infinite series because the series is obviously going to converge and it's a geometic progression. This makes the task pretty simple. I also took an x out because it's a common factor.

We also know that S (sum of infinite geometric series) = a/(1-r), where a is the first term in the series and r is the ratio between any two terms in a geometric series).

Therefore f(x) = x*(1/6 - 7^(-x)/6)

Put the numbers in and it works pretty nice. I've missed out a load of explanation, but there's your general solution.

[dave]

Incidentally, I've moved the thread over the number theory forum which is more appropriate for this.

[spacemanspiff]

that's why i hate mensa too. the question is misleading. I totally would have gone with 697/7 with my interpretation of the question.

[dave]

Also, notice that after x is bigger than about 5 or so, the term 7^(-x) is so small you can just ignore it. From my calculator, 7^(-20) = 1/79792266297612001, which is pretty small. It also managed to work out your answer exactly which was quite impressive, but it's so large that I couldn't be bothered typing it out.

(I have a TI-89, which is just the best calculator ever. I used it to check my series.)

[Roark]

Cool. I don't understand anything after f(x)= ... but cool!

I'll have to wait till my self-training catches up with your answer but thank you.

[dave]

Originally posted by spacemanspiff
that's why i hate mensa too. the question is misleading. I totally would have gone with 697/7 with my interpretation of the question.

I find the questions incredibly annoying, tedious and of no real use whatsoever to be honest. At the end of the day, it might broaden your knowledge of being able to interpret a question and perhaps a bit of lateral thinking, but it won't really help your mathematics in an extreme way.

[Roark]

Don't sweat wanting to just divide 697 by 7 to get the answer. I abbreviated the question from the book. Had I put it in verbatim, you wouldn't have gone there.

-Peter

[Radical Edward]

Originally posted by spacemanspiff
that's why i hate mensa too. the question is misleading. I totally would have gone with 697/7 with my interpretation of the question.

that is why mensa is mensa, because they would think of these sorts of things :P if the solution was easy, it wouldn't be mensa!

[JaKiri]

I like these questions, but that's an easier example of the variety :|

IMO opinion anyway.

<RadicalEdward> :P

[Radical Edward]

<MrL_JaKiri> 'I hate these lateral thinking puzzles, you have to think laterally' I think sums it up rather well