Puzzles with numbers and things

[davis_greatest]

Deadline for round 99 will be 12 noon GMT this Saturday, 6 January.

You'll like this

Gordon has got lots of boxes, and Charlie has got loads of balls. Gordon likes cubes, and Charlie likes triangles.

Now, Gordon's boxes are all different sizes - in fact, he has one box of almost every size imaginable (but all perfect cubes). His smallest box is just large enough to hold one snooker ball. Every next box he has is one ball's width wider than the previous box. So his 2nd smallest box is two balls' wide (i.e. it can hold 2x2x2 = 8 snooker balls); his next box can hold 3x3x3 = 27 snooker balls etc.

Charlie lays his red balls out in a triangle and then Oliver, who has lots of golden balls, does an exchange. Oliver will give Charlie golden balls in exchange for each of Charlie's red balls - and the number of golden balls that Oliver will offer for each red ball is equal to the number of red balls in Charlie's triangle!

For example, if Charlie's triangle contains 15 red balls, then Oliver will offer Charlie 15 golden balls for each red ball, so Charlie would end up with 15 x 15 = 225 golden balls!

Well, they play this merry game, and then Charlie puts his newly-acquired golden balls into Gordon's boxes - starting by filling the smallest box, then the next smallest etc, until all the golden balls are in boxes. It turns out that all of Gordon's boxes are filled completely - there is just enough space for all the golden balls!

Now, I forgot to mention - Charlie did not start with a triangle of 5 rows (15 reds). In fact, his triangle had over a million rows! Hehe

If Charlie's initial triangle had had one row more than it did, and they had played this game, then how many extra boxes would Gordon have needed?

Answers initially by Private Message please

[snookersfun]

Congratulations, snookersfun and robert602, who both found that the answer to round 98 is 825. (robert602 was after the deadline but his answer is still being accepted)

This can be found by adding every second triangular number, up to the 20th.

The triangular numbers are:
1
1+2 = 3
1+2+3 = 6
1+2+3+4 = 10
1+2+3+4+5 =15
1+2+3+4+5+6 = 21
.....
1+2+3+...+19+20 = 210

If we add every second one we get 3 + 10 + 21 + ... + 210 = 825.

snookersfun will, I hope, paste a nice picture of triangles to explain why this is the solution. (abextra, where are you with your smiley triangle pictures for this round? )

For n rows, we can find a nice formula: for n even, the formula is n(n+2)(2n+5)/24, which, if we put n=20, gives the 825 above.

Here is that picture, it lost some of its resolution during work in paint und uploading it, but it should do.


triangles of base 20-11 are not possible, 1+2=3 triangles of base 10 are possible, 1+2+3+4= 10 of base 9, ... upto 210 of base 1. Base 10 and 9 triangles are illustrated here, with a yellow triangle surrounding the possible balls that lie at the apex of the triangles of a possible size.

[davis_greatest]

Congratulations, snookersfun and robert602

You'll like this

Gordon has got lots of boxes, and Charlie has got loads of balls. Gordon likes cubes, and Charlie likes triangles.

Now, Gordon's boxes are all different sizes - in fact, he has one box of almost every size imaginable (but all perfect cubes). His smallest box is just large enough to hold one snooker ball. Every next box he has is one ball's width wider than the previous box. So his 2nd smallest box is two balls' wide (i.e. it can hold 2x2x2 = 8 snooker balls); his next box can hold 3x3x3 = 27 snooker balls etc.

Charlie lays his red balls out in a triangle and then Oliver, who has lots of golden balls, does an exchange. Oliver will give Charlie golden balls in exchange for each of Charlie's red balls - and the number of golden balls that Oliver will offer for each red ball is equal to the number of red balls in Charlie's triangle!

For example, if Charlie's triangle contains 15 red balls, then Oliver will offer Charlie 15 golden balls for each red ball, so Charlie would end up with 15 x 15 = 225 golden balls!

Well, they play this merry game, and then Charlie puts his newly-acquired golden balls into Gordon's boxes - starting by filling the smallest box, then the next smallest etc, until all the golden balls are in boxes. It turns out that all of Gordon's boxes are filled completely - there is just enough space for all the golden balls!

Now, I forgot to mention - Charlie did not start with a triangle of 5 rows (15 reds). In fact, his triangle had over a million rows! Hehe

If Charlie's initial triangle had had one row more than it did, and they had played this game, then how many extra boxes would Gordon have needed?

Answers initially by Private Message please

Congratulations to snookersfun and robert602 who both found that the answer is that one extra box would be needed by Gordon to house all the balls.

The number of boxes that Gordon needs is always equal to the number of rows in Charlie's triangle - whether it is one, or more than a million, or anything else! So adding one more row will mean that one more box will be required! The bit telling you that Charlie's triangle had over a million rows was, of course, a little red herring that I put there to amuse you.

The reason for the answer is that, if there are n rows in the triangle, then the number of golden balls is equal to the square of the nth triangular number - and the square of the nth triangular number is equal to the sum of the first n cubes...

[davis_greatest]

SO HERE IS THE SCOREBOARD AFTER ROUND 99

snookersfun.........................50 (halfway to that magic banana)
abextra...............................31
davis_greatest.....................24½
Vidas..................................12½
chasmmi..............................12½
elvaago...............................11½
robert602.............................10
Sarmu..................................8
The Statman.........................5
austrian_girl and her dad.........3½
Semih_Sayginer.....................2½
Snooker Rocks! .....................2½
Ginger_Freak.........................2½
April Madness........................1

[abextra]

Congratulations, snookersfun!

[davis_greatest]

My apologies to abextra for not having seen her correct answers to rounds 98 and 99 until this evening! So, double congratulations to abextra! And I have decided that davis_greatest should get that centenary point for round 100!

SO HERE IS THE SCOREBOARD AFTER ROUND 100

snookersfun.........................50 (halfway to that magic banana)
abextra...............................33 (nearly a third of the way to that magic banana)
davis_greatest.....................25½ (a quarter of the way to that magic banana)
Vidas..................................12½ (an eighth of the way to that magic banana)
chasmmi..............................12½ (an eighth of the way to that magic banana)
elvaago...............................11½
robert602.............................10
Sarmu..................................8
The Statman.........................5
austrian_girl and her dad.........3½
Semih_Sayginer.....................2½
Snooker Rocks! .....................2½
Ginger_Freak.........................2½
April Madness........................1 (a hell of a long way from that magic banana)

[snookersfun]

Yay, halfway to a banana.
100 rounds as well, we need a celebration

[elvaago]

davis_greatest, your answer to round 100 was wrong. So you don't get the point. :-)

As stated in the puzzle, my monkey stops when the colour is red. So the chance of the colour being red when he stops is not one in a hundred, it's 100%!

When it turns to red, my monkey will stop pressing the button.

[snookersfun]

Elvaago, you are a grinch!
looks like you had both possible answers here:

one in a hundred (assuming that he would have stopped for the nap after that time regardless of the colour of the light)

Edit: and assuming that by "done pressing the buttons" you mean done before the nap, and not on his return after the nap (otherwise the chance would be certainty)

[elvaago]

I don't want davis_greatest to reach the magic banana before I do! I said the round was worth zero points to begin with! He's a cheater! :-D

[snookersfun]

...and that would be the best strategy to get to that banana???lol

[davis_greatest]

davis_greatest, your answer to round 100 was wrong. So you don't get the point. :-)

As stated in the puzzle, my monkey stops when the colour is red. So the chance of the colour being red when he stops is not one in a hundred, it's 100%!

When it turns to red, my monkey will stop pressing the button.

Yes, "will stop", not "did stop" - hence my double answer to cover the ambiguity .... if you ever get that magic banana, I'm going to ask Oliver to "pass" it to me first, and we'll see how long it takes to make its way to you Maybe that will be question 200...

[davis_greatest]

Yay, halfway to a banana.
100 rounds as well, we need a celebration

Wow - impressive! Where did you find that? Did you happen to have it in your My Documents\My Pictures\Pictures of half a banana folder?

[davis_greatest]

Wow - impressive! Where did you find that? Did you happen to have it in your My Documents\My Pictures\Pictures of half a banana folder?

I've also just happened to notice that there are exactly 100 balloons! Even more impressive! So on your C: drive you have a My Documents\My Pictures\Pictures of half a banana\Pictures of half a banana with balloons\80-100 balloons\100 balloons folder?

[snookersfun]

Wow - impressive! Where did you find that? Did you happen to have it in your My Documents\My Pictures\Pictures of half a banana folder?

I made it (copious amount of copying and pasting into paint) and then put it into my My Documents\My Pictures\Pictures of half a banana + 100 balloons folder. Had a real hard day at work, as you can see