Puzzles with numbers and things

[davis_greatest]

Let's hear the joke! [Edit: I've now found the joke on the Jokes thread. ]

I've clarified in the Cakes question that Oliver is told that there is more than one cake in the box and each cake weighs more than 1 gram!

[davis_greatest]

Any answers should be sent by PM, but if anyone has any questions or requests for clarification, please post them on this thread so everyone gets the same information in response. I have been asked whether the weights could be 900, 900, 90, 90, 9, 9, 1, 1 gram. The answer, of course, is "no", because Oliver was told that the weights are equal.

[davis_greatest]

Round 82 Cream cakes - hint from Gordon

Gordon, who has been watching the events of Oliver's party intently, has decided to give you all a little hint, since the attempts to answer this question so far seem to have got a little stuck.

Gordon says this to you:

"Oliver is very clever, and may still be able to work out the weight of the cakes even if they do not weigh the same as an exact number of weights. For example, suppose that Oliver had already worked out that the total weight of the cakes might be 637g or 785g or 829g (I have made these numbers up), and that davis_greatest had given Oliver twenty weights, so each weight is 100g. Then if Oliver balanced the cakes against seven weights and found the cakes to be heavier, but balanced them against eight weights and found the cakes to be lighter, then Oliver would know that the cakes weigh between 700g and 800g, so must total 785g."


By the way, Gordon is clever too. He found a tiny slit in the cake box, which no one else has found. Here is a picture of him trying to look into the box, to see how many cakes there are.

[davis_greatest]

The deadline to round 80 has passed. Congratulations again to abextra and snookersfun, whose points have already been added, who found that the area of garden is triangular (an equilateral triangle) and covered with 81 triangular paving stones - 45 orange ones and 36 brown ones. Can anyone do a picture of it?

A couple of correct answers received so far to round 79. It is, of course, still open.


At the same time, you can have a go at....

Round 80: Ape Garden

I have given Charlie, Oliver and Gordon some parts of my garden for them to play. Charlie has his own area, whereas Oliver and Gordon wanted to share an area, so they jointly share a patch bigger than Charlie’s.

Gordon and Oliver’s area is a regular shape, and they decide to decorate it. So they both bring some paving stones, each the same shape as their area of the garden (but obviously much smaller). Gordon brings brown paving stones, the colour of his fur, while Oliver brings orange ones. The paving stones are all the same size, but Oliver brings more than Gordon.

Anyway, they lay the stones out, and they exactly cover their area of garden (with no gaps and no stones overlapping). In fact, they have laid them out cleverly, so that no two stones of the same colour touch (except perhaps at their corners, but no two stones of the same colour lie along a common edge).

Well, when they have finished, Charlie comes and has a look, but he says he doesn’t like it. So Gordon and Oliver pick up their paving stones, and rearrange them, again covering their area of the garden. This time, they place them so that every orange stone has at least one of its corners on the outside edge of Gordon and Oliver’s area of my garden, while no brown stone has any corners on the outside edge.

How many paving stones are there in Oliver and Gordon's area of my garden?

Answers by Private Message please. You can have until Initial Deadline of 21:00 GMT, Friday 8 December.

[snookersfun]

The deadline to round 80 has passed. Congratulations again to abextra and snookersfun, whose points have already been added, who found that the area of garden is triangular (an equilateral triangle) and covered with 81 triangular paving stones - 45 orange ones and 36 brown ones. Can anyone do a picture of it?

here is a picture...

[davis_greatest]

Thanks. Very nice How do you do these pictures?

[davis_greatest]

Round 80 solution

With the aid of snookersfun's picture, we see that a triangular garden area with n rows of triangular stones has n^2 stones in total.

For the first arrangement, with no orange stone and brown stone sharing a common edge, and more orange stones than brown, we have that:

n(n+1)/2 stones are orange (colour of Oliver's fur)
n(n-1)/2 stones are brown (colour of Gordon's fur).

For the final arrangement, with no brown stones having a corner lying on the outside of the garden area, there is a brown triangle in the centre with n-3 rows (n>=4).

So the brown triangle consists of (n-3)^2 brown stones.


Making the numbers of brown stones equal, before and after they have been rearranged, we have

n(n-1)/2 = (n-3)^2
So n^2 - n = 2(n^2 - 6n +9) = 2n^2 - 12n + 18
which gives
n^2 - 11n + 18 = 0
(n-2)(n-9) = 0


Since n>=4, we get n=9, giving n^2 = 81 stones in total.


You can't find a solution if you try to use square paving stones.

[davis_greatest]

Round 82 deadline extension

Oliver has had one correct answer so far to Round 82 - Cream cakes (which took a little while to get there ) and is going to extend the Initial Deadline to a Final Deadline of 23:00 GMT on Thursday 14 December...

Round 83 will come soon.

Oliver, my pet orang-utan, has decided he wants to have a party this afternoon and wants to invite some friends. So I go out to the baker, and come back and present Oliver with a nice, big, white, sealed box.

"What's in it?" asks Oliver, since he can't see inside.

"Cakes," I tell him.

"Oooh, " he says. "I like cakes. How many?"

"Well," I reply. "More than one! Each cake weighs the same whole number of grams (more than 1 gram!), they are all identical, and the box itself weighs virtually nothing!"

"Ooo oooh" says Oliver. "Interesting. But I didn't ask how much they weigh. I asked how many cakes are in the box. I want to know how many friends I can invite to my party!"

"Well, little Oliver," I reply to my pet ape, "if I told you the total weight of the cakes, you'd be able to work out for yourself how many there are!"

"Really?" he asks. "So what is the total weight of the box?"

"I'll tell you what," I say, "I'll give you a balance and some equal weights, with the weights totalling 2kg, and based on what I've already told you, you'll be able to use these to work out the weight of the box with cakes to work out how many cakes there are inside!"

"OK," says Oliver. So I place the weights on the table, and start to get up to fetch the balance for him.

Before I manage even to get up, Oliver looks at the weights, counts them, and without touching a thing, he says "Ah, great! Now I can invite all my friends, and there'll be just enough cakes for everyone to have one each!"

And so it turns out. How many are at Oliver's party this afternoon?


Answers by Private Message please. Initial Deadline will be 22:00 GMT on Monday 11 December.

Also don't forget that Round 80, Ape Garden, is still open!

[davis_greatest]

Scoreboard update

Round 82 - Cream Cakes has expired, so congratulations to snookersfun for her point for being the only one to determine that there were 37 at Oliver's party, each having one of the 37 cream cakes inside the box that I gave him.

snookersfun, would you like to post your explanation?

Half a point is being awarded to abextra, for getting halfway through the problem.

Congratulations also to davis_greatest, to whom elvaago has awarded an additional half-point for the cyclists problem.

SO HERE IS THE SCOREBOARD AFTER ROUND 82, BUT AWAITING POINTS FOR CHASMMI'S ROUND 77 (COUNTING LETTERS)

snookersfun……………………….…..40½
abextra...............................23
davis_greatest.....................17½
Vidas..................................12½
elvaago...............................8
chasmmi..............................8
robert602.............................6
Sarmu..................................6
The Statman……………………...……5
Semih_Sayginer.....................2½
austrian_girl and her dad.........2½
April Madness........................1

[elvaago]

Round... I always forget. 83?

Here's a neat one. Complete the following five series of numbers. Answers and explanations by PM only please! Some of them are easy, some of them are hard. If you get all 5, you get a point, if you get 4, you get half a point, 3 or
less, sorry! If no one gets them all before Monday 7 PM CET, I get a point. :-)
1) 1 - 8 - 9 - 16 - 17 - ???
2) 1 - 4 - 9 - 16 - 25 - ???
3) 1 - 2 - 3 - 5 - 8 - ???
4) 1 - 2 - 3 - 7 - 16 - ???
5) 3 - 3 - 1 - 2 - 4 - ???

[snookersfun]

Round 82 - Cream Cakes has expired, so congratulations to snookersfun for her point for being the only one to determine that there were 37 at Oliver's party, each having one of the 37 cream cakes inside the box that I gave him.

snookersfun, would you like to post your explanation?

Here is the explanation:
The statement of "if I told you the total weight of the cakes, you'd be able to work out for yourself how many there are!" means that the factorization of the total weight must be unique (e.g. prime1 * prime2) and, in the absence of any possible assumption about #of cakes or their weight, even Prime^2.

There aren't that many possibilities until a total weight of 2000 g:
primes of 2,3,5,7,11,13,17,19,23,29,31,37,41,43 yielding weights of 4,9,25,49,121,169,289,361,529,841,961,1369,1681, 1849g
Possibilities for equal weights adding up to 2000g are:
1,2,4,5,8,10,16,20,25,40,50,80,100,125,200,250,400 ,500,1000,2000g.

Now one basically has to find a weight range leading to one unique solution (i.e. only one of the possible weights is in that specific range and no ranges leading to several unique solutions as well). Using weights of 400 or 500g and the knowledge that those are sufficient for Oliver to know the total weight of the box, leads to 37, as 1369 is the only possible solution between 1000 and 1500g or 1200-1600g?

[davis_greatest]

That's right. There may have been either 4 weights (of 500g each) or 5 weights (of 400g each). Oliver would be able to tell that the total weight of the box of cakes is in the range 1000-1500 or 1200-1600, which must mean it is 1369 (37 squared). If the weight of the cakes had been anything else, Oliver wouldn't have been able to find the weight with only 4 or 5 weights. The fact that he could work out the weight of the cakes without even using the weights - just by knowing how many there were - meant that this must be the only solution.

If there had been 6 or more weights, Oliver would also have been able to use them to find the weight of the cakes if it had been something other than 1369g. Since he was able to discount there being more than one possible solution without even having to use the weights, there must be just the single solution above.

[chasmmi]

I will award a point to Davis and Sarmu as I have no clue and they did both go to an effort to give me an answer.

Sorry again.

[davis_greatest]

SO HERE IS THE SCOREBOARD AFTER ROUND 82

snookersfun……………………….…..40½
abextra...............................23
davis_greatest.....................18½
Vidas..................................12½
elvaago...............................8
chasmmi..............................8
Sarmu.................................7
robert602.............................6
The Statman……………………...……5
Semih_Sayginer.....................2½
austrian_girl and her dad.........2½
April Madness........................1

Sarmu perhaps a little fortunate there , but he did start late so missed most of the earlier rounds.

[davis_greatest]

Round 84 - I can't believe he's conceded the match

This round can run at the same time as elvaago's round 83...

Ronald and Michael are playing an exhibition snooker match. It is scheduled to go on forever, and they will continue playing as long as they are having fun.

However, both players are apt to concede at inopportune times and if either player ever falls two frames behind his opponent, he will concede the match and storm out of the arena, much to the disappointment of the spectators.

Ronald is twice as likely as Michael to win any given frame (so Ronald has a chance of 2/3 of winning each frame, and Michael has a chance of 1/3). Each frame is independent of every other.

What is the chance that Ronald will concede the match?

Answers to be posted on this thread, please - not by Private Message for this particular question.