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Three baskets and some balls are given to you. You are supposed to put these balls into these baskets by writing some positive numbers on the balls. But:

  • The numbers you are supposed to write on them have to be positive integer number,
  • In each basket, the numbered balls have to be distinct. (That also means you can put the same number in different baskets actually.)
  • The sum of the numbers on the balls in all baskets has to be at most $31$.
  • You do not have to put balls in every basket.

What is the maximum value of the multiplication of the numbers written on the balls?

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  • $\begingroup$ The rule: "You may not put any ball in any basket" is confusing, can you clarify this? $\endgroup$ – Green Oct 11 '17 at 19:43
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    $\begingroup$ To restate the problem: you are to maximize the product of a series of integers such that: (1) all integers are positive; (2) their sum is <= 31; (3) no integer may appear in the series more than 3 times. $\endgroup$ – Prune Oct 12 '17 at 0:06
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To make the product of the ball values as large as possible one needs to split up the 'budget' of $31$ into as many pieces as possible greater than $1$. Why $> 1$? Because balls with $1$ on them contribute to the sum but don't increase the product. Why as many as possible? Because $2 \cdot N \ge 2 + N$ (for $N \ge 2$), i.e. the total product will increase more by an additional factor of $2$ than by one of the factors being increased by $2$.
Edit: With regard to ffao's comment, I should add that there is exactly 1 case where the maximum number of factors is not the optimum, namely $2 \cdot 2 \cdot 2 < 3 \cdot 3$. In the final solution one will see that further 'finetuning' by replacing three $2$s by two $3$s is not possible, because there are already three $3$s (one for each basket) used.

So let's start with filling $2$s into each basket. The product is $8$ so far and the sum is $6$. Continuing with $3$s in each basket we have a product of $216$ and a sum of $15$. Next would be 3 $4$s, resulting in a product of $13824$ and a sum of $27$. But as the rest 'budget' to fill up to $31$ is just $4$, this can not be used to write another number on a ball. Instead one needs to use this to increase the existing numbers.

This increase should be spread among as many numbers as possible because $\left( N_1 + 1 \right) \left( N_2 + 1 \right) > N_1 \left( N_2 + 2 \right)$ for $N_1$, $N_2 \ge 2$. So let's first promote two $4$s to $5$s. Then we have a rest budget of $2$ from which we can promote a $2$ to $4$.

In total we then have in the baskets:
1. basket: $2, 3, 4, 5$
2. basket: $2, 3, 4, 5$
3. basket: $3$
Sum is exactly $31$ and product is $43200$.

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    $\begingroup$ "As many as possible" needs more justification. For example, 3*3 > 2*2*2. $\endgroup$ – ffao Oct 11 '17 at 21:29
  • $\begingroup$ @ffao Thank you for pointing that out. I have added this to my answer. $\endgroup$ – A. P. Oct 11 '17 at 21:54
  • $\begingroup$ The sum of the baskets shown is 36. Did you mean:$$(1): 3, 4, 5\\(2): 2, 3, 5\\(3): 2, 3, 4$$ That matches your description and has the indicated sum and product. $\endgroup$ – Paul Sinclair Oct 11 '17 at 23:07
  • $\begingroup$ @PaulSinclair You're right. One $5$ was too much. $\endgroup$ – A. P. Oct 12 '17 at 6:18

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