A Fistful of Brain Teasers

It would have been intriguing to have seen it as

X*Y = 55 and X + Y = 56

sjbabbey posted:

It would have been intriguing to have seen it as

X*Y = 55 and X + Y = 56

But which one would be one - x or y?

Innocent Bystander posted:

sjbabbey posted:

It would have been intriguing to have seen it as

X*Y = 55 and X + Y = 56

But which one would be one - x or y?

Certainly not -x 

since this boils down to solving a quadratic equation,  there are 2 pairs of solutions, (x, y),  as follows:

(11, 5) or (5, 11)

enjoy

ken

So, having combined the two equations to get y² - 16y + 55 = 0          (or the x² version)

Who then found the roots by using (y - a)(y - b) = 0

And found that the only two integer roots (5, 11) worked nicely

And who used the old "battling ram" formula x =[-b ±√(b² - 4ac)]/2a

Cheers

Don

And who knew instictively that 5*11= 55, and could ‘see’ in their head that those two added togethr made 16?

Innocent Bystander posted:

And who knew instictively that 5*11= 55, and could ‘see’ in their head that those two added togethr made 16?

 

Have to agree IB, can't see what need there is for all the above algebra / equations .......... the answer is staring one in't face!

dave marshall posted:

Innocent Bystander posted:

And who knew instictively that 5*11= 55, and could ‘see’ in their head that those two added togethr made 16?

 

Have to agree IB, can't see what need there is for all the above algebra / equations .......... the answer is staring one in't face!

That's why I added the following to the initial post...

PS when I saw this on another website, I thought "there's got to be more to this than meets the eye.....there wasn't"   

As you say, the answer was staring us in the face

I don't think you'll find this one "staring us in the face"    

How do I complete the following series: 1, 1/2, 1/4, 1/16, 1/25….......?

Ok, at first glance it looked  to be staring me  in the face, but it actually took me, quite a while to figure out....

Don Atkinson posted:

PS when I saw this on another website, I thought "there's got to be more to this than meets the eye.....there wasn't"   

What?! You haven't been entertaining us for all this time with examples from your 1963 Further Maths A-level textbook!!

Shocking!! Appalled!

Oh dear.............

My 1963 to 65 Maths A Level books do provide a significant source, but I will confess to :-

A few bits from the engineering maths I did at university

Bostock and Chandler (which I bought about 25 years ago to help my daughters) and

My memory of the weekly Brain Teasers that appeared in the Sunday Times between 1968 and 1973 when I had time to lay about in the sunshine in the Trucial States and Oman and

The occasional new one that I notice from time to time when browsing the local library, WH Smiths or (horror upon horror) the odd (often really odd) website.

Guilty as charged !

Don Atkinson posted:

I don't think you'll find this one "staring us in the face"    

How do I complete the following series: 1, 1/2, 1/4, 1/16, 1/25….......?

Ok, at first glance it looked  to be staring me  in the face, but it actually took me, quite a while to figure out....

6 is the next number. 1/2 squared is 1/4, 1/4 squared is 1/16, 1/16 squared is 1/256.

I had to ignore the starting 1, but the pattern amused me.

Hi EOINK, I'm so glad you added the smiling face after the 1/256. That is one of the more cheeky solutions that I have seen !!

Obviously a (very ?) nice try but.............

......as you have already accepted, you had to ignore the starting 1 to make it work.

“According to quantum theory, there is a probability that if I run toward a wall, I will pass through it due to tunneling effect. What's the equation to calculate this probability?”

My initial thought was P(E) = 0      where P(E) is the probability of the event being accomplished as stated.

However, the author of the above text approach the problem differently (and probably the wall as well !)

To be fair, his solution did generate an incredibly small probability, but it wasn't "0". However, it did show the probability increased as the running speed increased !

And to be even more fair, he did generate an equation for the life-threatening consequences of NOT passing through the wall and they approached "1" rather too fast for my liking !

Don Atkinson posted:

I don't think you'll find this one "staring us in the face"    

How do I complete the following series: 1, 1/2, 1/4, 1/16, 1/25….......?

Ok, at first glance it looked  to be staring me  in the face, but it actually took me, quite a while to figure out....

When I first saw this one, my initial thought was this series is formed by the formula 1/n². However, it doesn’t work for 1/2 and also the pattern is missing 1/9.

I then noticed that some of the denominators are the square of the preceding denominator eg 2² = 4, 4² = 16.  But 16² does not equal 25 and besides 1² is not equal to 2.

So I needed to find a more general rule that created the denominators; 1, 2, 4, 16, 25

Bingo ! ie here is one solution, but there are others…..

 (-11/12)x⁴ + (32/3)x³ - (487/12)x² + (371/6)x - 30

This generates the first 5 denominators given above and the next few denominators which are:

-4, -128, -426, -999 and -1970

Perhaps this next one is a bit easier than the 1, 1/2, 1/4, 1/16, 1/25 one above

and a bit more challenging than the 5, 11 one before that :-

How do you discover (or work out) the points of intersection between

y = x²   and   y = sin(x)

and what are your best estimates ?

I can only think of a Newton-Raphson iterative scheme right now...

enjoy

/ken

ken c posted:

I can only think of a Newton-Raphson iterative scheme right now...

enjoy

/ken

Well done ken,

That is certainly a good way and it will give very accurate answers, ken, assuming you have decent sin and cos tables and do enough iterations. I think about 5 or 6 gives a good answer if you start with a sensible first guess.

btw I don’t have any “elegant” solutions, but I can get good answers.

i’ll outline How I made my first “guess” after others have had an opportunity to respond.

with my good first guess, the R-N method only needed two or three itterations to give over 10 decimals consistency.

cheers

Don

Straight from the old A-Level maths book.....

What is the equation of the circle passing through the three points (1,2); (3,-4); and (5,-6) ?

Rather than the equation, you could just find the radius and co-ordinates of the centre.......

yes, things simplify quite a bit when you start from:

will take this further later as I am at work right now...

enjoy

/ken

brings back memories of school, a long, long  time ago...

enjoy

/ken

Neat solution ken.

my first (successful) solution was based on finding the intersection of the perpendicular bisectors of the two chords defined by the three given points. Crude, but worked !

certainly took me back a few years......Ok......decades !

Very sensitive, very political, but.........

If global population is currently 8bn and the Earth can only sustain 4bn (even with future technology).........

.......how long would it take to achieve this reduction if :-

          we limit children to two per heterosexual couple ( an heir and a spare)

          only 85% of such couples actually produce any children

          75 % of child-bearing couples choose to only have one child

          Infant and pre-adult mortality (combined)  is 10%

          generation gap is 25 years

          25% of the population is LGBT etc and don’t produce any children at all

assume :-

          male/female populations are of equal numbers etc

          assume life-expectancy remains constant.

          everybody lives in peace and harmony.

Any estimates ?

My first run through on the population reduction scenario suggests 56/57 years to halve the population.

I did assume life-expectancy to be 80 years and growth during each 25 year generation to be steady.