Puzzles with numbers and things

[davis_greatest]

Round 82 - Cream cakes

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]

The deadline for my puzzle with the two cyclists has passed. I got two answers, one right, one wrong. Congrats to davis_greatest for getting half a point.

Your explanation was very Mathematese. If you explain on this thread in easy to understand English, you get a full point!

Mathematese - I didn't use a single formula!

The cyclist in windy conditions takes longer, because he spends more time travelling slowly (when the wind is against him) than he spends travelling quickly (when the wind is behind him), so his average speed is lower than that of the cyclist who travels at a constant speed.

And in Mathematese - it is because the average speed travelled is the harmonic mean of speeds - and the harmonic mean is always less than or equal to the arithmetic mean (and so it is less than the average speed of the cyclist in non-windy conditions, whose speed is constant).

[davis_greatest]

The last to submit an answer to round 79, Bag of chalks, which has now closed, was chasmmi, and his point is added below. The answer to round 79 appears at the bottom this thread.

Sarmu seems to be getting close to joining abextra and snookersfun in solving Round 80, Ape Garden, but hasn't got there yet.

Also added is half a point to davis_greatest for round 81 about the cyclists. No doubt he will give his explanation in a bid to obtain the other half point, fairly shortly.

HERE IS THE SCOREBOARD AFTER ROUND 81, BUT AWAITING POINTS FOR CHASMMI'S ROUND 77 (COUNTING LETTERS) AND WITH ROUND 80 STILL BEING OPEN

snookersfun……………………….…..39½
abextra...............................22½
davis_greatest.....................17
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



Answer to round 79:

[davis_greatest]

one answer: 604,149,713

the other: 1,062,914,936

My answer was the first one. While I did it quite quickly and am by no means certain that it is spot on, I think that it is much more likely to be at or close to the mark than the other one. The second one, over 10 million numbers, would give an average length of 106.29 letters per number, which looks much too long to me. The first number gives an average of 60.41 letters.

If we look at a fairly typical / longish number, say
FIVE MILLION SIX HUNDRED AND SEVENTY TWO THOUSAND THREE HUNDRED AND TWENTY SIX,
that has 66 letters;

A shorter, but still not atypical number might be
TWO MILLION FIVE HUNDRED AND TWENTY EIGHT THOUSAND AND SEVENTY ONE
with 56 letters.

So I am more inclined to believe my calculation which gave an average closer to the 60 mark!

But I'm happy to hear the case for the other side!

[elvaago]

The deadline for my puzzle with the two cyclists has passed. I got two answers, one right, one wrong. Congrats to davis_greatest for getting half a point.

Your explanation was very Mathematese. If you explain on this thread in easy to understand English, you get a full point!

[chasmmi]

one answer: 604,149,713

the other: 1,062,914,936

[elvaago]

I fixed it.

[davis_greatest]

So Peter bikes the distance once with wind against him and once with the wind in his back.

I thought you said Peter lives in an area where it's never, ever windy?

[elvaago]

They are both exactly as quick if there is no wind.

[davis_greatest]

Who is the quicker cyclist in the absence of wind?

[elvaago]

Round 81.

Here's a quick and easy one for half a point only! PM only. Deadline Thursday 9 AM CET.

Tom and Peter are good friends. They love riding bikes. However, Tom lives in England, so where he lives it's always windy. In fact, it's always wind of the same exact speed, always coming from the west.

Peter lives in Australia, he lives in an area where it's never, ever windy.

Tom and Peter both begin on a bikeride. They use google maps to calculate a course that is exactly the same length in both countries. On the first day they ride one hundred miles going from east to west. On the second day they ride one hundred miles going back the other way. So Tom bikes the distance once with wind against him and once with the wind in his back. Peter bikes the distance with no wind whatsoever.

Now, tell me, who spent the most time cycling? Don't calculate it, but rather explain it to me in words why you think your answer is correct.

[davis_greatest]

OK - let's see them

[chasmmi]

Ok I have a confesion to make that I have lost my answer and can't even begin to remember what it was. I have recieved to vry different guesses at the answer and as I cannot say which is righti am inclined to just aard both with a point regrdless of correctness and then post both answers and see if we can conclude which is correct.

I is very sorry.

[davis_greatest]

Scoreboard updated - again!

Sarmu has now come in with a correct answer to round 79 (despite his first suggestion being that Gordon had just coloured the whole grid in black )

His point is added below, together with another point to abextra, who solved round 80 quickly last night, and one to snookersfun, who gave a correct answer to round 80 this morning.

SO HERE IS THE SCOREBOARD AFTER ROUND 78, BEFORE CHASMMI'S ROUND 77 (COUNTING LETTERS), AND WITH POINTS SCORED SO FAR ON ROUNDS 79 AND 80 (WHICH ARE STILL OPEN)

snookersfun……………………….…..39½
abextra...............................22½
davis_greatest.....................16½
Vidas..................................12½
elvaago...............................8
chasmmi..............................7
robert602.............................6
Sarmu..................................6
The Statman……………………...……5
Semih_Sayginer.....................2½
austrian_girl and her dad.........2½
April Madness........................1


I have just noticed that all the points add up to 129, which means that if we had all worked together and shared our points, we could have all had a banana to share 29 points ago!

[davis_greatest]

...and a point to abextra for round 79! The round is still open though!

SO HERE IS THE SCOREBOARD AFTER ROUND 78, BEFORE CHASMMI'S ROUND 77 (COUNTING LETTERS), AND WITH POINTS SCORED SO FAR ON ROUND 79 (WHICH IS STILL OPEN)

snookersfun……………………….…..38½
abextra...............................21½
davis_greatest.....................16½
Vidas..................................12½
elvaago...............................8
chasmmi..............................7
robert602.............................6
The Statman……………………...……5
Sarmu..................................5
Semih_Sayginer.....................2½
austrian_girl and her dad.........2½
April Madness........................1