Brain Teasers ? or 50 Years On........... ?

The insight that makes this easier is that you can toss out (or drink) some of the beer along the way.

Nice explanation sj..........especially the lie-down

JRHardee posted:

The insight that makes this easier is that you can toss out (or drink) some of the beer along the way.

 

You take a new job with a starting salary of £10,000 per annum. (ok, ok. feel free to add a Zero to all the numbers, including those below, if that makes the job worthwhile )

You are offered two ways in which your salary can increase.

Plan A: a £500 raise every six months – meaning that every six months, your salary for the next six months raises by £500.

Plan B: A £2,000 raise every year – meaning that every 12 months, you salary for the next 12 months raises by £2,000.

Which plan would you choose ?

Salary puzzle:-

Plan B is the better solution unless you are expecting to work for less than 15 months. Seems too easy for a "Don puzzle" (unless I'm missing something........!).

Steve D

Plan A is better as below

Total        6 mos        12 mos        18 mos        24 mos

Plan A        5,000        10,500        16,250        22,250

Plan B        5,000        10,000        16,000        22,000

Doh! The "something I've missed" is not reading the question properly.....

......your salary for the next six months raises by £500 (my workings were based on £500 per annum).

Steve D

Hm, looks like long term (after 30 months) Plan B is better

Total        6 mos        12 mos        18 mos        24 mos        30 mos        36 mos        42 mos        48 mos

Plan A        5,000        10,500         16,250        22,500        28,750        35,250        42,000        49,000
    
Plan B        5,000        10,000        16,000        22,000        29,000        36,000        44,000        52,000

Hi,

I get the same numbers as SJ for Plan B. For Plan A my numbers are:

5.000 -> 10.000 -> 17.000 -> 23.000 which is even better (maybe too good to be true).

I now agree that Plan A is better. My workings indicate that Plan A will always earn £500 more for each full year.

Steve D

Every year plan A brings in £500 more than plan B

Cumulative totals for every 6 months:

Plan A: 5000, 10500, 16500, 23000, 30000, 37500, 45500, 54000.

Plan B: 5000, 10000, 16000, 22000, 29000, 36000, 44000, 52000.

Mulberry posted:

Hi,

I get the same numbers as SJ for Plan B. For Plan A my numbers are:

5.000 -> 10.000 -> 17.000 -> 23.000 which is even better (maybe too good to be true).

Plan A without typing error : 5.000 -> 10.000 -> 16.500 -> 23.000

steved posted:

Salary puzzle:-

Plan B is the better solution unless you are expecting to work for less than 15 months. Seems too easy for a "Don puzzle" (unless I'm missing something........!).

Steve D

.....................

Plan A is always £500 better off, each and every year.

The "wording" of the two options is "Clear", in the sense that you are told very precisely how much money will be added to your starting salary. But the wording obviously makes the £2,000pa increase look so much more attractive.

I wonder how many statements made by politicians follow this sort of ploy ? Perhaps I should copy this one into the "Sleeping" thread !!!

Perhaps stating that you could have a salary increase of 10% of your initial starting salary every six months or 20% of the starting salary every year would be less misleading. Boils down to exactly the same thing, but somehow, makes you think more carefully, perhaps ?

Many thanks to all who tried and well done to those who got there (eventually) - it takes courage to publish your thoughts on a forum !

The picture above incorporates five sizes of circle. From smallest to largest there are :-

• Six yellow circles
• Seven red circles
• Three blue circles
• One dotted circle that just sits inside the triangle
• One large circle with a solid perimeter

Please accept the limitations of my Power-point drawing, the circles and triangle “just” touch each other tangentially. No “overlaps” no “slight” gaps.

If the diameter of the smallest circles (yellow) is 1.0 what are the diameters of each of the other circles ?

Don Atkinson posted:

Plan A is always £500 better off, each and every year.

The "wording" of the two options is "Clear", in the sense that you are told very precisely how much money will be added to your starting salary. But the wording obviously makes the £2,000pa increase look so much more attractive.

I wonder how many statements made by politicians follow this sort of ploy ? Perhaps I should copy this one into the "Sleeping" thread !!!

Perhaps stating that you could have a salary increase of 10% of your initial starting salary every six months or 20% of the starting salary every year would be less misleading. Boils down to exactly the same thing, but somehow, makes you think more carefully, perhaps ?

Many thanks to all who tried and well done to those who got there (eventually) - it takes courage to publish your thoughts on a forum !

Don,

 I don’t think this is correct.

 Plan A Year 1 (salary £10,000)

Earnings for the first 6 months are £5,000. After 6 months salary is increased to £10,500, which means earning in the second 6 months will be £5,250. Total earnings £10,250 in the first year.

 Plan A Year 2 (salary £11,000)

Earnings for the first 6 months are £5,500. After 6 months salary is increased to  £11,500, which means earning in the second 6 months will be £5,750.  Total earnings £11,250 in the second year.

fatcat posted:

Don Atkinson posted:

Plan A is always £500 better off, each and every year.

The "wording" of the two options is "Clear", in the sense that you are told very precisely how much money will be added to your starting salary. But the wording obviously makes the £2,000pa increase look so much more attractive.

I wonder how many statements made by politicians follow this sort of ploy ? Perhaps I should copy this one into the "Sleeping" thread !!!

Perhaps stating that you could have a salary increase of 10% of your initial starting salary every six months or 20% of the starting salary every year would be less misleading. Boils down to exactly the same thing, but somehow, makes you think more carefully, perhaps ?

Many thanks to all who tried and well done to those who got there (eventually) - it takes courage to publish your thoughts on a forum !

Don,

 I don’t think this is correct.

 Plan A Year 1 (salary £10,000)

Earnings for the first 6 months are £5,000. After 6 months salary is increased to £10,500, which means earning in the second 6 months will be £5,250. Total earnings £10,250 in the first year.

 

 Plan A Year 2 (salary £11,000)

Earnings for the first 6 months are £5,500. After 6 months salary is increased to  £11,500, which means earning in the second 6 months will be £5,750.  Total earnings £11,250 in the second year.

Oh Dear !

I was very careful to use the same words as I wrote down from this "teaser" a few years ago, especially the bit ".....meaning that every six months, your salary for the next six months raises by £500."

At that time, and also when I posted this above, those words seemed straight forward. They were intended to convey the message that your "wages" for the next six months increase by £500 ie as per my spreadsheet.

Just goes to show how an "honest politician" such as me can unintentionally get it so ambiguous as to be mis-leading !!!! (ie wrong !)

My sincere apologies to all those who tried this teaser and interpreted my careless wording differently to me, especially those who posted their solutions above and whom feel cheated. You all deserve real credit for your explanations, the validity of which I un-wittingly overlooked . And also a big thank you to Frank for pointing out my ambiguous (to say the least) wording.

The two pictures above show a few lines of inquiry that you might explore to generate enough equations for a solution.

Probably best to work in Radii rather than Diameters.

I have prepared my solution to the "18 circles and a triangle" problem.

But I am dreading Frank's review if I have goofed yet again...........

My solution………………well, the first part of it !

Label the circles from smallest to largest A, B, C, D and E and let them have corresponding radii a, b, c, d, and e.

In the first diagram I have shown a vertical line which represents the radius of circle D, the one with the dashed line. But it also represents the four radii of the smallest circle A, plus three radii of B. So I can write an equation :-

d = 4a + 3b      [1]

Likewise the other two thick black lines which are radii of E, generate equations :-

e = b + 2c        [2]

e = 4a + 5b      [3]

Now for the difficult bit…………..(I’ll leave that for a while…..)

....................

Now for the difficult bit…………..

The second diagram shows a Rhombus. You need to convince yourself (prove) that it really is a Rhombus and not just any old quadrilateral !!

Alternatively you could prove that the two triangles made bisecting the Rhombus are equilateral triangles.