Puzzles with numbers and things

[snookersfun]

Just a quick update, before leaving for a bit...:
Monique solved this one in no time... Congrats!
By Saturday answers can come up on the thread!

[snookersfun]

to make up for so far unsolved Round 256 (let's leave it open for a while longer, maybe a hint to follow by next week...)

Round 260: Minesweeper
DSCF1998.jpg

the regular fleet again, this time the numbers in grid point to ship parts adjacent to them (incl. diagonally). Ships can't sit on the numbers themselves.

[davis_greatest]

Yes, Monique (apart from the little typo: n^2 = 87025 and not 87024) - perfect and congratulations!

[Monique]

Building on R258 we know that for each day (#Rows-1)³ = 4 x Number of uneaten apples
Therefore at the end of the competition we have
4* total number of uneaten apples = 2^3*SUM( i^3; i=1 to (n-1)/2) = 2^3*[1/2 * (n+1)/2 * (n-1)/2]^2 where n is the number of rows on the last day of the competition. This takes into account the fact that the number of rows is always odd and starts at 3.

so (n^2 -1)^2 = 32*236661768
and n^2 = 87024 and n=295.
Because Barry starts with 3 rows and increases by 2 every day the #days is (n-1)/2 = 147.

[davis_greatest]

147 days ...

Congratulations Monique! That is the correct answer!

And the explanation?

[snookersfun]

147 days ...

oops, too late... and huge hints this time

Here is the solution to the previous one:

There are 201 rows in that triangle, as one receives an n of 100
after solving for number of apples:

basically twice the possible difference between max number of bananas and that won by each ape (∑2*3n²), minus the highest possible difference ((3n²)as one can't choose the 1 rows), minus the apples eaten (n):

∑2*3n² - 3n² -n= n(n+1)(2n+1)-3n²-n= 2n³ = 2.000.000

with n=(m-1)/2 and m=the number of rows in the big triangle

[Monique]

147 days ...

[April madness]

I think I fainted of the smell before entering that exhibition and I don't like apples that much anyway

but thanks for another nice question, surely I'll visit more often to see the answer

[davis_greatest]

For this next one, a little harder, you really need to know how to do round 258... so snookersfun, Monique and abextra will have a bit of an advantage!

But, to be fair, I should put the answer up to round 258 first. So, all you really need to know for round 258 is that one can write:

(Rows-1)³ = 4 x Number of uneaten apples = 4 x 2 million = 8,000,000
So number of rows = 201

Round 259 - Apple Museum

On day 1 of the Great Ape Snooker Championship, Barry opens his shop and gives away apples and bananas to Gordon and the chimps just as he did in round 258 - except that instead of having 201 rows of balls he has only 3 rows of balls in the triangle. The chimps all leave with their apples (eating some exactly as they did in round 258).

On day 2, Barry does the same, giving away bananas and apples, but he now has 5 rows of reds in the triangle.

Every day of the championship, he does the same, giving away bananas and apples, but adding 2 rows of reds each day.

So the number of rows of reds in Barry's triangle is 3 (on day 1), 5 (on day 2), 7, 9, 11, 13,.... each time an odd number.

On the last day of the championship, the chimps leave Barry's shop and put all their uneaten apples, 236,661,768 of them in fact, in Barry's museum. April Madness visits but doesn't touch them as most of them have gone rotten.

How long does the Great Ape Snooker Championship last?

Answers on the thread please.

Edit: Hint - the sum of the first n cubes is equal to the square of the n-th triangular number

[April madness]

I go to this thread like to a museum. not to touch anything

[davis_greatest]

abextra has also solved round 258! Congratulations!

Solutions on the thread please!

[davis_greatest]

... and Monique in with the 2nd!

[davis_greatest]

Round 258 update

snookersfun first in with a correct answer to this one...

[davis_greatest]

I'll try, snookersfun... as long as it stays open for a while ... First I'll add another round...

Round 258... Bary's Bigger Banana Giveaway

Barry the Baboon decided that the Banana Giveaway Promotion worked so well, that he decides to do it again on a bigger scale. Again, he lays out in the entrance of his shop a big equilateral triangle of red snooker balls (bigger than before), touching as they would in a pack in snooker, with an odd number of rows.

The promotional sign explains that any ape can come into the shop and choose any Lucky Banana Number that he wants (as long as he doesn't choose the same Lucky Banana Number that an earlier ape has already chosen). Every Lucky Banana Number must be 2 or greater.

Just like before, any ape coming into the shop will be given some bananas for every line of touching red balls, going in any direction... but, in order to count, every line of touching balls must have at least as many balls as the ape's nominated Lucky Banana Number! The number of bananas won for each line is the ape's nominated Lucky Banana Number.

For example, if an ape's Lucky Banana Number is 7, then the only lines that count are those with 7 or more balls, and 7 bananas will be won for each such line. Each line only counts once (e.g. a line of 3 balls cannot be counted as 2 lines of 2 balls).

First into the shop is my pet gorilla, Gordon. He inspects the triangle, and then, being very smart, chooses his Lucky Banana Number to get the maximum possible number of bananas.

After that, a series of chimpanzees come into the shop, one by one. Each chimpanzee chooses a Lucky Banana Number and wins some bananas. The chimpanzees each win fewer bananas than Gordon - so Barry, in order to keep his customers happy, gives each chimpanzee a number of apples too, to go with the bananas. In fact, every single ape (Gordon and each of the chimpanzees) wins the same total number of fruit (apples + bananas combined). Gordon's fruit, of course, consist solely of bananas.

Every time that a chimpanzee wins a number of bananas that is not the same as the number of bananas won by any other ape who entered the shop before him, he eats one of his apples! For example, if a chimpanzee wins 50 bananas and no other ape had won exactly 50 bananas, he will eat one of his apples.

This goes on, with chimpanzees coming in and winning bananas and apples, until it is not possible for any more bananas to be won (because there are no possible winning Lucky Banana Numbers left)!

With no more fruit on offer, Gordon and the chimpanzees leave the shop, carrying a grand total of 2 million uneaten apples!

And so... how many rows of red balls in Barry triangle?

Answers by Private Message initially please.

[snookersfun]

Round 256 update:

there is no update yet, or did I miss any input due to TSF problems?
d_g you might be interested in this one (dare you!)