Puzzles with numbers and things

[Monique]

Thanks for correcting me D_G... I also changed the wording a few times when elaborating the puzzle. I love very dark chocolate (at least 85%)

[davis_greatest]

This evening, I went to play snooker. However, it was a rather boring affair and I struggled to make any sizeable breaks, as all the balls were missing. Eventually, I found them - my three apes were playing with them and had shared them out.

They had, among them, the complete set of 22 balls, and each ape was adding up the value of his balls (1 for a red, 2 for a yellow, 3 for a green etc, with no value for the cue ball).

Although my pet gorilla Gordon had the fewest balls (and Oliver, my little pet orang-utan had the most), the value of Gordon's balls was at least 3 times the value of my pet chimpanzee Charlie's. Nevertheless, the value of Charlie's balls was at least as great as Oliver’s.

What balls did each ape have?

Congratulations to Monique and snookersfun who made short work of solving this early this morning - I'll leave it open a little while longer for the next person who wants to post the answer on this thread.

[Lee Vilenski]

Bah, I tried that one, and it looked easy. But, after 5 minutes, I got bored.

[icantplay]

Again apologies for being so ignorant. . . .
concerning the previous puzzle, how come the number of rows in the small triangle is a third the number of rows on 147147147147147????

And again . . .

If Oliver has 7 reds and the white, 7 points, and Charlie has 6 of the remaining reds and the yellow, 8 points, then Gordon would have the last two reds and the five remaining colours, 27 points?

[davis_greatest]

Again apologies for being so ignorant. . . .
concerning the previous puzzle, how come the number of rows in the small triangle is a third the number of rows on 147147147147147????

From post 1774 above, we have that:

(1) Rows in Chimpanzicus triangle = Rows in Olivero triangle + Rows in Gordonicus triangle

and from the information in the question

(2) Rows in Olivero triangle = 2 x Rows in Gordonicus triangle

Therefore, putting equation (2) into (1), we get

(3) Rows in Chimpanzicus triangle = 2 x Rows in Gordonicus triangle + Rows in Gordonicus triangle
= 3 x Rows in Gordonicus triangle

Therefore Rows in Gordonicus triangle = 1/3 x Rows in Chimpanzicus triangle
= 1/3 x 147147147147147


Or see snookersfun's post 1778...

And again . . .

If Oliver has 7 reds and the white, 7 points, and Charlie has 6 of the remaining reds and the yellow, 8 points, then Gordon would have the last two reds and the five remaining colours, 27 points?

Getting close - however, that doesn't quite work, because you were told that Gordon has the fewest balls, whereas in your answer both Gordon and Charlie have the same number of balls (7). Almost there though!

[davis_greatest]

I'll leave round 167 open a little while longer if anyone is still trying it. For those who found it too easy, we have...

Round 168 - Who's got the balls? II

(This is probably no harder to solve than round 167, and maybe it's easier - I don't know - but certainly it's harder for me to remember.)

Tonight, after my game of snooker, which this time went very well as all the balls were there, I let my three apes play with the balls again. Again, they shared among them the complete set of 22 balls, and each ape added up the value of his balls (1 for a red, 2 for a yellow, 3 for a green etc, with no value for the cue ball).

Charlie had over 3 times as many balls as Oliver; and the value of Charlie's balls was over twice that of the balls of Oliver and Gordon combined! Cheeky Charlie! (Mind you, Gordon's balls were of less value than Oliver's, but don't tell Gordon that.) The ape who ended up with the yellow ball had one red more than the ape with the brown ball, whereas the number of balls held by the ape with the black ball was less than 3 times the number held by the ape playing with the yellow ball. I do remember that the value of Gordon's balls was either greater or less than the number of balls he held, but I don't recall which.

Apart from that, I don't remember at all which ape had which balls... so, please help me - what balls did each ape have?

[davis_greatest]

... and, to go with rounds 167 and 168 (which are still open), here is

Round 169 - Thirteen triangled

So, round 169, which is 13 squared. I could try to ask something about 13 and squares, but here instead is a question about 13 and balls and triangles...

Take a set of 13 red snooker balls, and arrange them touching each other on the bed of a snooker table in any way you choose. Within the group of balls, there may be various triangles.

Within your group,
- for each triangle of 3 reds, you score 3 points;
- for each triangle (if any) of 6 reds, you score 6 points;
- for each triangle (if any) of 10 reds, you score 10 points.

The triangles must be equilateral and consist of touching reds, i.e. like the usual formation of balls within a pack of reds in snooker (except that the pack would be smaller).

Example:

OOOOOOO
.OOOOOO

has, I think, 11 different triangles of 3 reds (can you find them?), so would score a total of 33 points.

(Actually, I'm quite pleased with that arrangement - it looks pretty good - I perhaps shouldn't have given that away as an opening example!)


Arrange the balls to get the highest score you can - and bid here on this thread!

[The Statman]

I'll bid 79 points:

OOO
.OOO
..OOO
.OOOO

1 × 10 = 10
5 × ..6 = 30
13 × 3 = 39

[The Statman]

I'll bid 82 points:

...OO
..OOO
.OOOO
OOOO

1 × 10 = 10
5 × ..6 = 30
14 × 3 = 42

[The Statman]

I'll bid 92 points:

...O
OOOO
.OOO
OOOO
...O

2 × 10 = 20
6 × ..6 = 36
12 × 3 = 36[/QUOTE]

[Lee Vilenski]

Er....

---0
--000
-00000
00000
Er...

Thats 10x2
3x8
6x5

20+ 24 + 30 = 74... Not a bad effort!

[davis_greatest]

Er....

---0
--000
-0000
000000

Isn't that 14 balls, instead of 13?

[Lee Vilenski]

ok, changed, doesn't change the outcome.

[davis_greatest]

I'll bid 92 points:

...O
OOOO
.OOO
OOOO
...O

2 × 10 = 20
6 × ..6 = 36
12 × 3 = 36

A fine submission, The Statman! In fact, this was the pattern I had in mind originally when I asked you to use 13 balls. My instinct would be that this is the best possible - although I haven't thought about it yet to see if this can be proved, and would be delighted to see anyone better it!

[davis_greatest]

Er....

---0
--000
-00000
00000
Er...

Thats 10x2
3x8
6x5

20+ 24 + 30 = 74... Not a bad effort!

ok, changed, doesn't change the outcome.

Um.... haven't you now just moved one ball from one row to another from your original drawing - there are still 14 balls there!