View Full Version : One for the mathematicians

22nd Jan 2009, 22:14
While we were trying to find some mathematical challenges for a very bright 8 year old we looked at Pythagorean triangles. I listed integer triangles which were not multiples of any smaller triangles, ie discard 6, 8, 10 as it is just a multiple of 3, 4, 5. I think these are called primitive solutions. I wanted to see if there was an easy way to spot possible Pythagoras triangles without actually doing the calculations. The 8 year old does huge number multiplications in his head, far beyond what I can do, and we wanted to find a way to keep up with him!

The list goes like this,

3, 4, 5
5, 12, 13
8, 15, 17
7, 24, 25
20, 21, 29
12, 35, 37

and what I noticed is that the longest side (hypotenuse) increases in steps of 4 and 8 alternately eg 5, 13 ,17, 25, 29, 37, 41 etc

EXCEPT that there isn't one at 49. the next one follows the pattern and appears right on cue at 53 and the following ones are at 61 then 65, pattern returned.

Why is the pattern broken at 49?


22nd Jan 2009, 22:32
I think the answer is E=MC^2

22nd Jan 2009, 22:41
Ask the eight year old.

It might keep him out of mischief for a while.

But to be fair to him, make sure he has access to a supply of squared paper, so that he can explain it in pictures.

DX Wombat
22nd Jan 2009, 23:06
Why not contact MENSA (http://www.mensa.org.uk/)? They have a section for gifted and talented children (http://www.mensa.org.uk/mensa/gifted_and_talented_support.html)and their parents.

22nd Jan 2009, 23:30
Along the same lines, you could also try NAGTY, http://www.nagty.ac.uk/.

I was in that in a former life, yep I was :ok::p

22nd Jan 2009, 23:39
NAGTY (http://www.warwick.ac.uk/gifted/) :confused:

22nd Jan 2009, 23:45
and what I noticed is that the longest side (hypotenuse) increases in steps of 4 and 8 alternately eg 5, 13 ,17, 25, 29, 37, 41 etc

EXCEPT that there isn't one at 49

the next one follows the pattern

you've just demonstrated by direct observation that your supposed 'pattern' is bollocks as far as patterns go. You're not a 'climate scientist' are you?

22nd Jan 2009, 23:51
By coincidence, I have been asked by a customer to deliver the world's largest greetings card. This is some kind of stunt & the thing will be on show at the London marathon. Anyway, it's 9 metres long by 3 metres high.

We realised that to get moved by lorry, we would have to construct a triangular frame to support it and allow it to rest at an appropriate angle in order to lower it from 3 metres so that it can actually make the journey from the factory in Kent and actually make it under a bridge or two.

To work out the dimensions of the frame, we did the calculations using the Pythagoras theorem in converse, that is we already know the hypotenuse & one other side, which is the platform width of the trailer.

There we were just today reminding one another of how we sat in maths lessons at school moaning about bloody trianges & how we would never have to use them in the real world.:rolleyes:

2.2 metres if anyone's still awake.

23rd Jan 2009, 00:04
you've just demonstrated by direct observation that your supposed 'pattern' is bollocks as far as patterns go. You're not a 'climate scientist' are you?

Heh, I have to agree with BlooMoo.

For kicks I even wrote an algorithm to check it out, so can confirm your pattern does no hold true.

If you want to know how to build a matrix in Excel of integer Pythagoras triangles let me know and I can share the formula.

Beatriz Fontana
23rd Jan 2009, 08:46
I note that the maths boffins have come up with a formula for beer goggles (http://news.bbc.co.uk/1/hi/4468884.stm). I wonder how much practical research was carried out?

23rd Jan 2009, 13:21
an easy way to spot possible Pythagoras triangles

What's a Pythagoras triangle? Do you mean a right-angle triangle? You can easily tell one of these because it will always have one angle which is 90deg. Teach normally gives her 8yo the value of at least one of the angles to make it easier. Unless they're breeding them smarter these days??? But listening to the kids on their mobile phones or behind the McDonalds counter I don't think this is the case...

using the Pythagoras theorem

Pythagoras' theorem or the Pythagorean Theorem.

23rd Jan 2009, 13:33
Have you ever kissed a girl?:rolleyes:

23rd Jan 2009, 16:20
Dear me. Plenty of people shooting down the notion that there was a pattern that was somehow broken. They are quite right, of course, but they did not give the reason why there is the anomaly at 49.

It's all to do with the facts that 49 is 7 X 7 and the square root of 2401. Now do you see it?

Captain Stable
23rd Jan 2009, 16:58
There's a joke about the squaw on the hippopotamus...

Thanks, I've already got a coat.

23rd Jan 2009, 17:01

Continue with the pattern until you reach the next anomaly.

All should be clear by then.


23rd Jan 2009, 18:08

23rd Jan 2009, 18:47
Dr. Ron Knott at the Univ. of Surrey has a most entertaining website on recreational maths. His site has more than you will ever want to know about Pythagorean triangles, triple and patterns.


23rd Jan 2009, 18:49
Good one Gordy! :D

23rd Jan 2009, 21:20
BlooMoo, I am not a climate scientist and I don't get excited by the threat of global warming as the earth has, in general, been getting warmer since the last ice age and it has actually been warmer than it is now at least once since then.

Birrddog, I generated the numbers on a programmable calculator and I thought that maybe I had a bug in my programme when the pattern failed. This suspicion was largely because the programme ran FIRST time, never wrote a bug free programme with that many lines before.

Farrell, the next anomaly occurs at 65 when you find that there are two solutions, 77 is another missing one and then 85 has two solutions. Still makes no sense to me.

It looks as if Nagty need a bright 8 year old to sort out their website!


23rd Jan 2009, 21:42
The 8 year old does huge number multiplications in his head, far beyond what I can do

Some autistic people can calculate like that. Could he be on the autistic spectrum?

23rd Jan 2009, 21:47
r6a, imho I don't think it is possible to pick up a pattern such that you are looking for; as you pointed out, there are multiple combination's that can generate a particular integer hypotenuse, and with very small lengths on say x axis vs. large numbers on y axis (or vice versa).

Took me a while to write my code... very rusty :O

23rd Jan 2009, 23:03
Weren't any such things as 'calculators' when I was at school. All 'mental' until we moved-on to log-tables. Didn't use slide-rules until university.

DX Wombat
23rd Jan 2009, 23:18
Could he be on the autistic spectrum?Oh for heaven's sake! :mad: What is the matter with you? Just because a child is gifted does NOT mean he or she is on the Autistic Spectrum. There are some who are, but there are many who aren't and there are some with very serious learning difficulties but it would be equally wrong to suggest that all those with learning difficulties are autistic.

24th Jan 2009, 02:03
I suspected that some Real Mathematicians would have looked at this Pythagorean Triple problem at some point, and so it is: have a look at what the makers of Mathematica have to say on the topic, here (http://mathworld.wolfram.com/PythagoreanTriple.html). There's plenty there to keep you all busy... for example, did you know that there are four triangles with a hypotenuse of 65? (Only two are primitive, however, the other two are multiples.)
65 = 16+63 = 25+60 = 33+56 = 39+52.

26th Jan 2009, 10:42
"Autistic Spectrum"

If Pink Floyd ever bring out a new album.....

29th Jan 2009, 00:26
I sent the text of the original 'problem' as outlined in p*st #1 (why (no) 49?) to my son (a Cambridge double-first mathmo now working in telecoms).
This is his solution:-
I created a spreadsheet to figure out some more primitive pythagorean triangles - this is as far as I got:-
A B C Diff
3 4 5
5 12 13 8
8 15 17 4
7 24 25 8
20 21 29 4
12 35 37 8
9 40 41 4
28 45 53 12
11 60 61 8
16 63 65 4
33 56 65 0
48 55 73 8
13 84 85 12
36 77 85 0
39 80 89 4
65 72 97 8
20 99 101 4
60 91 109 8
15 112 113 4
44 117 125 12
88 105 137 12
17 144 145 8
24 143 145 0
51 140 149 4

So the short answer is that 49 is not the only exception to the rule. In fact, there are lots of exceptions to the rule, so you can't really call it a rule.

However, there is SOME kind of pattern here. Every value in column C is in the format 1+4x, for some value of x.

So the answer is, 49 can't be the longest side of a pythagorean triple, because its prime factors (i.e. 7) are not in the format 1+4n.

Hope this helps . . .