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We shall also close round 366 then. The answer - and congratulations to those who found it - is one, four, seven.
The crux to solving it is to note that:
(i) If Charlie's "rectangle" (which we shall see in a moment must turn out to be a square) has x rows and y columns, then the bottom layer of Oliver's triangle has (x+y) rows.
(ii) Charlie's two triangles then have (x+y)^2 balls. So xy must divide (x+y)^2.
(iii) If xy divides (x+y)^2, then x=y, so Charlie has a square, and Oliver has 4 times as many balls as Charlie.
(iv) Hence there are 4 crates of red balls - and this we know without even considering the total number of balls there are, nor have we yet considered Gordon. Also the number of balls in each crate is x^2 = y^2, hence a square.
(v) Then look at the highest square that divides 147 million to find how many balls in a crate. Our first try - 49 million - is too big, so look at the next highest square, by dividing by 4. Et voila.
Round 368 to follow....
The crux to solving it is to note that:
(i) If Charlie's "rectangle" (which we shall see in a moment must turn out to be a square) has x rows and y columns, then the bottom layer of Oliver's triangle has (x+y) rows.
(ii) Charlie's two triangles then have (x+y)^2 balls. So xy must divide (x+y)^2.
(iii) If xy divides (x+y)^2, then x=y, so Charlie has a square, and Oliver has 4 times as many balls as Charlie.
(iv) Hence there are 4 crates of red balls - and this we know without even considering the total number of balls there are, nor have we yet considered Gordon. Also the number of balls in each crate is x^2 = y^2, hence a square.
(v) Then look at the highest square that divides 147 million to find how many balls in a crate. Our first try - 49 million - is too big, so look at the next highest square, by dividing by 4. Et voila.
Round 368 to follow....
Originally Posted by davis_greatest
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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…
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…
Again, we'll leave the round open a little longer. :snooker:
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