# Today's puzzle



## John-H (Jul 13, 2005)

In the Footprints Cafe each table has three legs, each chair has four legs and all the customers and the three members of staff have two legs each. There are four chairs at each table. At a certain time, three-quarters of the chairs are occupied by customers and there are 206 legs altogether in the cafe.

How many chairs does the cafe have?
(Must show working)


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## leopard (May 1, 2015)

I reckon 32 chairs.

200 feet if you don't include the waiters

Take a quarter off = 8 tables worth...

Which makes 32...


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## SamDorey (Dec 31, 2016)

after putting so many numbers in my calculator I've lost track with what I was working out :lol:

My answer is 19

Edited, meant 19 not 14.


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## John-H (Jul 13, 2005)

leopard said:


> I reckon 32 chairs.
> 
> 200 feet if you don't include the waiters
> 
> ...


You got the first bit about ignoring the six legs belonging to the waiters.

3/4 x 200 = 150 though. Or did you mean 4/3 x 200 = 266.666?

Eight tables worth of legs (if occupied fully by customers) must be 8 x (3 + (4x(4+2))) = 216

How did you work out eight tables?



SamDorey said:


> after putting so many numbers in my calculator I've lost track with what I was working out :lol:
> 
> My answer is 14


Not the right answer I'm afraid. It's a head scratcher! I thought of writing it into an equation but it's easy to get confused between the total number of occupied and unoccupied chair legs and your variable meanings being total or individual etc.


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## leopard (May 1, 2015)

No,I divided the 200/25 to give 8 which gives the amount of chairs× legs of 32 ? [smiley=bomb.gif]


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## John-H (Jul 13, 2005)

SamDorey said:


> after putting so many numbers in my calculator I've lost track with what I was working out :lol:
> 
> My answer is 19
> 
> Edited, meant 19 not 14.


No, not 19 either - and you need to show how you got there :wink:


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## leopard (May 1, 2015)

So is it 32 ?


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## SamDorey (Dec 31, 2016)

leopard said:


> No,I divided the 200/25 to give 8 which gives the amount of chairs× legs of 32 ? [smiley=bomb.gif]


200 is the 3/4 amount and 266 is 4/4


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## leopard (May 1, 2015)

"A quarter off which is 8 table's worth" makes sense to me but that was a quick mental calculation which might confuse others,though I still stand by 32...


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## SamDorey (Dec 31, 2016)

I'm starting to think you're right. May I change my answer? :lol:


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## BW57 (Jun 26, 2017)

Let x = no of chairs, so no. of legs(chairs) = 4x

No of tables = x/4, so no. of legs(tables) = 3x/4

No of customers = 3x/4, so no. of legs(customers) = 6x/4

Ignoring the staff legs, the sum of the above legs = 200. i.e. 4x + 3x/4 + 6x/4 = 200

Multiply all the terms by 4 to get rid of the fractions gives. 16x + 3x + 6x = 800 leading to 25x = 800 so x = 32

i.e. 32 chairs.


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## Roller Skate (May 18, 2015)

I'm going for 32 ... I'm using "odds" as my proof of work. :lol:


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## brittan (May 18, 2007)

BW57 said:


> Let x = no of chairs, so no. of legs(chairs) = 4x
> 
> No of tables = x/4, so no. of legs(tables) = 3x/4
> 
> ...


This is correct.

I started with Let No of tables = x

Similar working gives 3x + 16x + 6x = 200 so x = 8

Hence No of chairs = 4x = 32


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## John-H (Jul 13, 2005)

Congrats to everyone who had a go and was brave enough to post their answer - it was a tricky one! Thanks for working it out too Brian.



BW57 said:


> Let x = no of chairs, so no. of legs(chairs) = 4x
> 
> No of tables = x/4, so no. of legs(tables) = 3x/4
> 
> ...


Correct! You have the correct answer and first showed fully how you got there 

Leopard - you had the correct answer but I couldn't quite see the full reasoning and apologies for not saying that the answer was correct but I didn't want to stop people thinking about it.

I'd worked out if Tt = total tables and Ct = total chairs, then ignoring waiters, customer, table, chair legs must be:

(3/4 x Ct x 2) + (3 x Tt) + (4 x Ct) = 200

As Tt = Ct/4 you can re-write:

Ct x ((3/4 x 2) + 3/4 + 4) = 200

So, Ct = 200/((3/4 x 2) + 3/4 + 4) = 32 chairs.

Same thing.

A clue, by the way, was in the title "Today" as it came from BBC Radio 4's Today programme puzzle series. Their quoted answer is:



> The answer is 32 chairs. You may have approached this in different ways, but here is one explanation.
> 
> 206 legs minus the 6 legs of the staff = 200 legs. On average, at each table, there are 3 table legs, 16 chair legs and 6 customer legs which gives a total of 25 legs per table. 200 divided by 25 is 8, hence there are 8 tables. As there are 4 chairs at each table the answer is therefore 32 chairs.


Which maybe what Leopard was thinking.

I think I need a lie down now ... [smiley=dizzy2.gif]


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## leopard (May 1, 2015)

John-H said:


> Congrats to everyone who had a go and was brave enough to post their answer - it was a tricky one! Thanks for working it out too Brian.
> 
> 
> 
> ...


That's the one and I'll humbly accept the prize at the monthly Admin and moderator's fine dining soiree


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