Puzzles with numbers and things

[snookersfun]

If this is the maximum, the quiz meister is yet to pronounce , why does it take so long to find? It looks easy when the sequence is complete in front of you.

I am positive d_g used telepathic suggestion on us.43.gif Must be somewhere in those lengthy rules...

[moglet]

Hadn't thought of that

I shall have to go for some counselling, the stress, the pressure, not coping.......

[davis_greatest]

Some good bids here - well done! (Not that I have checked them, but am relying on you to check those of each other )

To confirm though, 197 is not the best possible!

Edit - checked moglet's 197 now - and yes, it is correct and very good, but not a maximum

[davis_greatest]

I'll leave round 365 open for someone to bid higher!

But also post:

Round 366 Barry's Crazed Crate Banana Bonanza

Charlie, Oliver and Gordon go to buy some balls from Barry the Baboon’s Ball Shop. Charlie buys a crate of pink balls, Oliver some crates of red balls and Gordon some crates of black balls – each crate contains the same number, and together they bring back 147 million balls to my house! And some bananas. Quite a sight!

Charlie lines his pink balls out in a grid formation, using them all – lining up rows and columns evenly in straight lines.

Oliver makes a great big triangular pack of red balls, and then places a second (slightly smaller) triangular pack of red balls on top - so the top level balls fill all the hollows made by the balls in the lower level. Oliver finds that he exactly uses all his balls to do this.

Then naughty Gordon steals the top row of Charlie’s balls, and then steals the end column from what is then left in Charlie’s rectangle. Gordon finds that the number of pink balls he has stolen is exactly the same as the number of rows in the top layer of Oliver’s triangle!

Oliver had bought the maximum possible number of crates consistent with the above information.

Gordon had bought the minimum possible number of crates consistent with the above information.

How many crates did (i) Charlie, (ii) Oliver, (iii) Gordon bring back to my house?

(And for a bonus point - what is amazing about this answer? )

Answers initially by Private Message…

[snookersfun]

Some good bids here - well done! (Not that I have checked them, but am relying on you to check those of each other )

To confirm though, 197 is not the best possible!

Edit - checked moglet's 197 now - and yes, it is correct and very good, but not a maximum

right, 198 now

[moglet]

Yes, 198, blind as a bat, could have gone straight to it yesterday

[moglet]

R365 update

I'm certain Snookersfun saw this and a sight quicker than I did

Black/black 1+14
Blue/blue 1+10
Green/green 1+6
Pink/yellow 1+6
Black/black 1+14
Blue/blue 1+10
Green/green 1+6
Pink/yellow 1+6
Black/black 1+14
Blue/blue 1+10
Green/green 1+6
Black/yellow 1+7
Pink/Pink 1+12
Blue/yellow 1+5
Black/black 1+14
=155
Yellow/yellow 4
Green/green 6
Brown/brown 8
Blue/green 5
Pink/yellow 6
Black/black 14
=43

[snookersfun]

I'm certain Snookersfun saw this and a sight quicker than I did

yep, that's it!
Small consolation for never seeing the 197 yesterday

[J T]

You got me on this one and it was supposed to be easy!!:

[moglet]

Welcome J T, it is a great thread that will drive you crackers, but if you are looking for an occasional workout for the old grey cells, you could do worse.

[davis_greatest]

right, 198 now

I'm certain Snookersfun saw this and a sight quicker than I did

Black/black 1+14
Blue/blue 1+10
Green/green 1+6
Pink/yellow 1+6
Black/black 1+14
Blue/blue 1+10
Green/green 1+6
Pink/yellow 1+6
Black/black 1+14
Blue/blue 1+10
Green/green 1+6
Black/yellow 1+7
Pink/Pink 1+12
Blue/yellow 1+5
Black/black 1+14
=155
Yellow/yellow 4
Green/green 6
Brown/brown 8
Blue/green 5
Pink/yellow 6
Black/black 14
=43

Congratulations! Yes, this is it - 198 - and it is the maximum.

Meanwhile, on round 366, snookersfun has found the answer and explained why it is amazing. :snooker: Any other takers? Or is a little hint wanted?

[J T]

Thanks moglet for the welcome, i think my old grey cells are giving up the ghost!

[moglet]

Thanks moglet for the welcome, i think my old grey cells are giving up the ghost!

Yeah, it is exhausting, mine (grey cells) have other things to do but I can't resist a challenge even if I can't always solve the problems set. I hope one day to set a problem myself that will confound all the contributors here. Well, I can live in hope

[davis_greatest]

Welcome J T.

Meanwhile, moglet is there with a correct answer for round 366! (A few gaps in the explanation to close, but the numbers are right! )

I'll put up a few hints shortly to help close the gaps for those still trying!

[davis_greatest]

Right, some hints then to round 366 .... in hidden text, so just select the text to look at the hints if you want them!

Hint 1:
(
Ignore Gordon’s crates at first. First work out how many crates Oliver must have. )


Hint 2: Find the red herring
(
“Oliver had bought the maximum possible number of crates consistent with the above information.” is a red herring. Why? )


Hint 3:
(
It is a red herring because there can only ever be one possibility for the number of crates Oliver had – and you can work that out even without needing to know the total number of balls there are! What is the only possibility? )


Hint 4: Simple expression for number of balls Charlie has

(
Suppose Charlie’s grid has x rows and y columns. Then it has xy balls (OK – not much of a hint there!) So number of balls in a crate = xy )


Hint 5: How many balls does Gordon steal?

(
First he takes a row of y balls. The he takes a further ?-? balls from one of columns left.

So the number of rows in the top layer of Oliver’s triangle is y+?-? and the number of rows in the bottom layer is ?+? = n( say) )


Hint 6: What is the form of the number of red balls of Oliver?

(
Take a triangular pack with n rows and then take another slightly smaller pack with n-1 rows. Move the balls in each triangle a little so that each triangle becomes right-angled – now you can fit the two triangles together to form a _QU_R_ with ? rows.

Or show algebraically that n(n+1)/2 + n(n-1)/2 = … )


Hint 7: finding the number of crates for Oliver (part 1)

(
So Oliver has n^2 = (x+y)^2 balls which must divide the number of that Charlie had in his one crate.

If (x+y)^2 is divisible by xy, what does this tell you about x and y? )



Hint 8: finding the number of crates for Oliver (part 2)

(
Divide x and y by any common factors, to get x’ and y’
Then we know (x’+y’)^2 is divisible by x’y’
So x’^2 + y’^2 is divisible by x’y’, and hence also by x’
Since the first term x’^2 is divisible by x’, y’^2 must be divisible by x’. So y’ is divisible by x’, as they have no common factors.

Similarly, by symmetry, x’ is divisible by y’, so x’=y’ (=1).

Hence x=y and Charlie’s grid is a _QU_R_! )


Hint 9:

(
You should now be able to work out the number of crates Gordon has, and the number in each crate is a square.)

Hint 10 : Working out how many crates Gordon has (part 1)

(
If the number of crates Gordon has is as small as possible, the number of balls in each crate must be as ----- as possible! )


Hint 11 : Working out how many crates Gordon has (part 2)

(
What are the factors of 147 million? What is the highest square that divides 147 million? If that is too big, what is the next biggest square that divides 147 million?)