Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ π -¹ ² ³ °

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I had a look and I liked it. Whilst the "area under the curve" is good, the idea of "multiplication for things that vary" could help some who can't apply concepts away from the way they are taught. I also suspect that it does provide the "ah" moment.

Thanks for showing me that.

**random_fruit**- Replies: 3

You have probably come across the KenKen game by now.

*Tedious explanation: a square grid left empty except for part rows, part columns, sub-squares, etc, so that every square of the grid is 'covered' by one part-row, part-column, etc. Each part-row, part-column, sub-square, etc., has in its top-left-most square a result and a simple arithmetic operator, one of add, subtract, multiply, divide. The subtract and divide part-rows and part-columns can only be two cells big. Also, each complete row and complete column must contain all the digits starting at 1 and upto the size of the grid, but in any order. More complicated shapes, such as letter L are allowed. The puzzle is to deduce the values in each blank cell so that all the arithmetic is satisfied and each row and column contains 1 to grid-size.*

Here is one which you can only understand in a mono-space font: (cut-n-paste it)

```
+---+---+---+---+
|6* |12* |
+ +---+---+---+
| |6+ |3+ |
+ + +---+---+
| | |3 |5+ |
+---+---+---+ +
|9+ | |
+---+---+---+---+
```

*(Mono-spaced that for you.)*

which you can try to solve for yourselves. (Answer below...)

What I was wondering, having solved several of these now, is how many possible combinations of layouts of digits are possible, ignoring the cells and arithmetic. Placing numbers 1-n in a row obviously gives factorial n, written n! Placing numbers in a grid (so that each row and column contains 1 to n exactly once) has more combinations.

A one-sided grid has one combination. A two sided grid has two combinations. (They are:

12

21

and

21

12)

I think the number of possible combinations, N, is given by:

N = (n!) * (n!)

for n=3 and n=4. Can anyone develop (or quote) a generalised proof, where n>=2? Or am I simply wrong on this?

And the answer to the little grid above is

2143

3412

1234

4321

and I hope that I haven't made a silly in drawing up and checking the puzzle works!

I see someone has edited the post I submitted on 2009-01-02. I had written:

_s_e_x_ over here = sine: opposite / hypotenuse

but someone or something had removed the naughty word _s_e_x_ and replaced it with gender giving the nonsense you can see above.

Thanks for the answers - so it seems if one tried to define some "new maths" involving ln(-1) it would turn out that sqrt(-1) could be defined using the "new" value ln(-1).

Yes, I did know that e^iπ = -1

Thanks for both of the replies, and I'll go away and have a ponder on this.

**random_fruit**- Replies: 4

So, the square root of minus 1 is called i and gives us imaginary then complex numbers. Another function that is not defined for negative numbers (or zero, for that matter) is logarithm. Has anyone ever used log(-1) or perhaps ln(-1) to extend maths? If so, does anyone have a URL so I can read about it? (and then get confused!)

Thanks,

random_fruit

Thank you both. The words "algebraically closed" tell me what I needed to know, but I hadn't made the leap that rules out 3D. when I have some time to spare I think I had better look Quaternions up in Wiki and hope I can understand some of that.

random fruit

**random_fruit**- Replies: 3

One could say the real number line is one dimensional. Then, one could say that imaginary numbers give us a second dimension, and complex numbers allow us to 'explore' the plane. So, what piece of maths allows us to 'move' into a three-dimensional space?

I know we could just give 3 real numbers (x,y,z) which describe a point in 3D. I could for myself work out 3 separate lines/planes/solids using 3 sets of one-dimensional co-ordinates. But, that NOT what I want. What I wish is an idea as 'clever' or as 'complicated' or even as 'simple' as imaginary numbers which takes us from two to three dimensions.

Has this ever been done? I so, I have yet to hear of it. Mind you, that's not to say either way... 'cos my maths education pretty much stopped at age 18 before university. But I am curious that as we live in a 3D world I have not yet come across a 3D maths.

Can an answer be put in 'simple' terms, such as I can recall from 1975 when I finished 'A' level maths in the UK? Thanks,

random_fruit

Perhaps in the time it takes you to refresh, the number of users goes down? It's probable that the database system (MySQL) only has a user connected during the period the messages are being selected from the database. This is only happening between a user clicking on a message title and the message text appearing on his/her screen.

Amusing... when I pressed the 'submit' button for this post, I got the same message myself!

I'd just like to point out that in post #5, where Ricky divides by product of a and b, his transform does not work if either a or b is zero. Therefore, his super-complex formula does not work for either y = ax (because b=0) or y = b (cos a=0).

Just me being picky.

I'd like to see the idea expressed by Paul Lockhart taken into English schools. Every week I work with electricians who cannot recall Ohm's law (V = IR) and who cannot transpose it (I = V/R) and then who cannot use it to predict the current (I) given the voltage (V = 230) and a resistance (R = 0.35). When I tell them to divide 230 by 0.35 they cannot operate the calculator in front of me to get the arithmetic result (go on, try for yourself... did you get 657?) This matters for electrical safety!

And so very often those who left school at 16 in England can't do this simple piece of maths, and think its nasty and evil and what they became an electrician to avoid. If Paul Lockhard's approach will overcome this depressing mess, I'd think it wonderful, and I feel his point of view should be given a chance to make a difference.

So, if electrons and other sub-atomics have free-will, does this mean we need Isaac Asimov's notion that a sufficiently large number of unpredictable entities can be predicted accurately, as in the Foundation Series? (see http://en.wikipedia.org/wiki/The_Foundation_Series) Perhaps if we use this notion, can we avoid being considered merely the sum of our atoms.

I wouldn't like that outcome, because I quite like the thought that I have my own free will.

Is this getting close to the stuff that ought to be in Dark Discussions at Cafe Infinity?

Thanks for posting this, Laterally. I really enjoyed the engrish site.

I agree with Jane to some extent, but I also think that the Times has it right when it reports that "Maths just isn't cool." Until we get over our anti-science bias, we will continue to have our children and young adults looking away from maths and science.

Ten to fifteen years ago, before Maggie Thatcher became PM, I heard an interview on BBC Radio 4 "Today" programme (that's a weekday breakfast-time news and current affairs programme on the radio) where someone said "The trouble with the UK is that it's a business and industry country with an anti-business bias." Maggie made it OK to be keen on business, and gave us the yuppie 80s and 90s. What we need today (if my thought is correct) is a similar "Maths is cool, Science is cool" movement in the UK.

Anyone agree?

One of my children (all now grown into sensible adults, but that's another story) finally understood percentages when I talked with her about it in terms of money.

First, one percent of one pound is one penny. (This is £ sterling in UK, but applies just as well to Euros and cents, Dollars and cents, etc).

Second, a one percent increase applies to every pound of wages earned in a week. So, for examples:

£1/week + 1% gives £1.01/week

£2/week + 1% gives £2.02/week

£1/week + 2% gives £1.02/week

£2/week + 2% gives £2.02/week

We discussed what percentage her pocket-money should go up next birthday in similar fashion. She soon got the idea from the pattern above.

Finally we discussed reductions by a percentage. At a sale in a shoe shop, some shoes were advertised as 25% off. I said, it probably meant that those shoes now cost 75p for each £1 they used to cost before the sale. We did some examples:

£1 gone down to 75p

£2 gone down to £1.50

£3 gone down to £2.25

£10 gone down to £7.50

£40 gone down to £30.

On the last she said "Huh, I can tell how often you buy womens' shoes" but I protested "But Dads don't buy womens' shoes."

A few days later she followed up with: Why did you say "it probably means"? I joked that some shop-keepers with a particularly keen desire to emphasise the value-for-money of their goods, could compute the percent reduction another way. I said, if a pair of shoes now cost £50 and used to cost £100, could that be computed as 100%? She said "no" but I reminded here that the reduction (£100-£50) was £50 which was clearly reduction-in-price/original-price = £50/£50 and therefore 100%. She said "that's cheating" and I said it was not beyond the guile of some. I'd like to think that UK trading standards departments have an opinion about this, but I have no experience.

Finally, whenever I hear a percentage being mentioned on the news: "house prices in the UK have fallen 12% over the last year, says" I have three questions: (a) show me the input data (b) how do you justify the computations and (c) who wishes to prove what with it.

In my opinion, percentages are often poorly understood by many people and this results in a mis-use of them as heresay and opinion, and not useful arithmetic. I think it would be a good idea if only interest rates on loans and savings, and rates of tax, were expressed in percentages.

Hey, that's my education bit (and rant) over, hope some of you enjoyed it.

The reason why brainf*ck is such a nasty programming language is that it doesn't use assembler style nmemonics. A very long time ago I learnt a language called "CESIL" which is documented on the web. See http://www.obelisk.demon.co.uk/cesil/ The power of the languages is similar, but the programmer-hostile form of the programs means it is really hard to read.

Computer science has a similar beast named "Turing Machine" and a theory (theorm?) which says that if a program can be written it can be written for a Turing Machine. Look here: http://en.wikipedia.org/wiki/Turing_machine for some more details.

As a sage put it, there's nothing new under the sun.

One that I was taught at a Electrician's class, consisting mostly of teenagers, for the sin, cos and tan:

gender Over Here Can Always Happen To Old Age

which means:

gender Over Here: Sin = Opp / Hypot

Can Always Happen: Cos = Adjac / Hypot

To Old Age: Tangent = Opp / Adjac