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wowbagger
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Three sons (generalised)
« on: Aug 21st, 2003, 6:06am » |
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A father has three sons of ages x, y and (x+y). Now, x and y happen to be the first and second digits of the father's age (a two-digit number), respectively. Furthermore, the sum of all ages, i.e. father + x + y + (x+y), equals 45.
Find all possible tuples (x, y, b) that solve this problem in base b. This means that all numbers (including the 45) are written in base b.
Hint: Look at the equation you arrive at for particular values of x.
This is an extension of the riddle posted by Kelvin in "Children in a family". Take a look at towr's answer (for base 10) if you don't know how to solve such problems.
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| « Last Edit: Aug 21st, 2003, 6:13am by wowbagger » |
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towr
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Re: Three sons (generalised)
« Reply #1 on: Aug 21st, 2003, 6:19am » |
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partial
1 <= x < b, 0 <= y < b, x + y + (x+y) + (x*b+y) = 4*b + 5, b >= 6
(b+2)x +3 y = 4*b + 5
x = (4*b + 5 - 3*y)/(b+2)
x = 4 - (3*y+3)/(b+2)
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wowbagger
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Re: Three sons (generalised)
« Reply #2 on: Aug 21st, 2003, 6:31am » |
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Your last line is useful for another bound on x, towr (I arrived at that bound by simple trial).
Regarding possible values of b, I suggest exploiting the equation in your second (hidden) line.
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towr
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Re: Three sons (generalised)
« Reply #3 on: Aug 21st, 2003, 6:43am » |
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yes I know.. but I had to watch tv (Yu-gi-oh) 
continuation
1<=x<=3
y = (4*b + 5 - (b+2)x)/3
y = b+1, (2b+1)/3 , (b-1)/3
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| « Last Edit: Aug 21st, 2003, 6:47am by towr » |
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wowbagger
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Re: Three sons (generalised)
« Reply #5 on: Aug 21st, 2003, 7:00am » |
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on Aug 21st, 2003, 6:48am, Sir Col wrote:
I've already corrected it. Thanks anyway.
As for your example: (x, y) = (3, 2) works in base 7, but (1, 5) definitely doesn't. I guess you just miscalculated x for y = 5.
Not sure whether I should leave this in now that you've removed that part of your post. Maybe it can serve as an example for others.
What I'd like you to come up with is a general form (x, y, b). Not surprisingly (?), there are infinitely many such tuples.
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| « Last Edit: Aug 21st, 2003, 7:04am by wowbagger » |
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towr
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Re: Three sons (generalised)
« Reply #6 on: Aug 21st, 2003, 7:17am » |
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::
x=2, y = (2b+1)/3 OR x=3, y = (b-1)/3
b = 3n +1
(x,y,b) = (2, 2n+1, 3n+1) or (3,n,3n+1) [forall]n>1[in][bbn]
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| « Last Edit: Aug 21st, 2003, 7:27am by towr » |
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Sir Col
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Re: Three sons (generalised)
« Reply #7 on: Aug 21st, 2003, 7:21am » |
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Oops! For b=7, I meant y=2,x=3 and y=5,x=2; you're right I miscalculated x. 
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wowbagger
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Re: Three sons (generalised)
« Reply #8 on: Aug 21st, 2003, 8:29am » |
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Your answers are the same I got, towr. Only I wrote them in a form with n[in][bbn]: { (2, 2n+3, 3n+4); (3, n+1, 3n+4) } .
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towr
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Re: Three sons (generalised)
« Reply #9 on: Aug 21st, 2003, 8:38am » |
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hmm.. yes, but that would raise the age-old question is 0[in][bbn]
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Sir Col
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Re: Three sons (generalised)
« Reply #10 on: Aug 21st, 2003, 10:52am » |
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What's with zero? I would say, an emphatic, no. Natural numbers are called as such, because they are countable. It is quite natural to have 1 chair, 2 chairs, et cetera. You cannot have 0.5 chairs, as a chair remains a chair until it is not a chair – this is the principle of function/utility. Equally, and sorry to make reference to the 'other' thread, you cannot have 0 chairs. It is an utter nonsense to talk about zero anythings.
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towr
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Re: Three sons (generalised)
« Reply #11 on: Aug 21st, 2003, 11:00am » |
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no it's not.. You can easily have zero chairs. And you might as well look up mathworlds page on natural numbers, and see that indeed the 'community' doesn't agree wether or not zero should be included.
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Sir Col
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Re: Three sons (generalised)
« Reply #12 on: Aug 21st, 2003, 11:15am » |
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This is precisely why it took so long to incorporate zero into mathematics. The ancient Greeks presented well founded reservations about the confusion that would arise from both quantity (count) and quality (function). The Greeks felt that zero, literally, made no difference in counting: a+0=a; and so was redundant in arithmetic. In terms of function, it was meaningless to talk about the function of nothing: zero lots of something is absolutely equilvalent to any amounts of nothing. For example, 0 chairs = 25 nothings (= nothing); their function is equivalent.
The use of zero in mathematics should be entirely reserved for the absence of something – namely, quantity. It is necessary to make numbers, like 10 (1 ten and 0 units), meaningful.
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towr
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Re: Three sons (generalised)
« Reply #13 on: Aug 21st, 2003, 12:32pm » |
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on Aug 21st, 2003, 11:15am, Sir Col wrote:| The use of zero in mathematics should be entirely reserved for the absence of something – namely, quantity. It is necessary to make numbers, like 10 (1 ten and 0 units), meaningful. |
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Frankly I don't see what you have a problem with..
A chair is something. Zero chairs thus signifies the absence of something. Namely the absence of any chair.
Zero chairs does however not signify the absence of tables, doors, forks, spoons or sporks. 'Nothing' on the other hand does.
zero is the absence of something
nothing is the absence of anything
each has their own function.
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| « Last Edit: Aug 21st, 2003, 12:34pm by towr » |
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Sir Col
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Re: Three sons (generalised)
« Reply #14 on: Aug 21st, 2003, 4:15pm » |
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That is precisely my point. The absence of something in a mathematical context, and that would include counting, allows for the use of zero; and although correct, it is redundant.
If I asked you how many chairs are in a particular room, and upon surveying the room you see a rectangular arrangement of 3 by 4 chairs, would you seriously reply, "There are 3 lots of 4 chairs," or would you reply, "There are 12 chairs?"
In the same way, if I asked the same question regarding an empty room, would you reply, "There are 0 chairs," or, "There are none?"
Although I would understand both, and both are correct, it is inefficient to say more than required.
Colloquially anyone would accept both, and dare I admit that most people would prefer to say, "There are 0 chairs." However, from a logical perspective, the use of zero in this context can at best be justified as a negation of expectation and, therefore, is not equivalent to a mathematical zero; it is simply used in place of not/none.
As an example, I am sure that you would be surprised if I randomly said, "There are zero bars of gold in my house." Unless you've managed, where Interpol has failed, to connect me with the famous Parisian gold bullion heist of 1997. Dang!
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Icarus
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Re: Three sons (generalised)
« Reply #15 on: Aug 21st, 2003, 4:45pm » |
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Given recent developments in the field of reproductive technology, you may have been too hasty in discarding the solutions (0, 4n+3, 3n+1) and (1, n+3, n). 
(Actually not - since the father's age is a two digit number and the sons are both single digits, but I couldn't resist.)
Concerning 0 and the Natural numbers [bbn]: while philosophical nitpicking does go on about the status of 0, in practice whether or not a mathetician considers 0 [in] [bbn] almost always comes down to what is most convenient at the time. I have many times seen the same mathematician use both definitions (on different topics, of course). In fact, I am one of those mathematicians!
For reasons beyond rational explanation, mathematicians prefer the name "natural numbers" to "whole numbers". Therefore the use of the Whole numbers, [smiley=bbw.gif], when 0 is needed has never caught on. Indeed, while William has provided [ bbn ] as a short-cut for [bbn], he considered [smiley=bbw.gif] not to deserve a shortform. To get it you have to use [ smiley=bbw.gif ].
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towr
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Re: Three sons (generalised)
« Reply #16 on: Aug 22nd, 2003, 1:06am » |
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on Aug 21st, 2003, 4:15pm, Sir Col wrote:| If I asked you how many chairs are in a particular room, and upon surveying the room you see a rectangular arrangement of 3 by 4 chairs, would you seriously reply, "There are 3 lots of 4 chairs," or would you reply, "There are 12 chairs?" |
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I might see that there were three rows of four, if I thought the configuration important. It gives extra information, and wether that information is redundant depends on the context.
Quote:| In the same way, if I asked the same question regarding an empty room, would you reply, "There are 0 chairs," or, "There are none?" |
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'none' isn't a numbver, so it's not logically an answer to the question 'how many' which clearly asks for a count, a number.
Not that logic or lack of it ever stopped anyone 
Quote:| Although I would understand both, and both are correct, it is inefficient to say more than required. |
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Wether you say zero or none, both are 4 letters, so neither is more efficient than the other. It's also not necessareily inefficient to say more than required, in most cases it's actually preferred socially. If you always give very short (but correct) answers you sound very unfriendly.
Quote:| As an example, I am sure that you would be surprised if I randomly said, "There are zero bars of gold in my house." |
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No more surprised than if you said "there are no bars of gold at my house". Random phrases in themselves imply surprise..
I still don't quite see what you object to.
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