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The problem from a math magazine that I am struggling to solve.

Little Red Riding Hood has 60 cookies and 5 guests. She needs to split the cookies on the plates in a way that everyone has the equal amount of pies. But she does not know how many guests are going to come. That's why she decided to split pies on the plates so that it will be able to split the cookies between 2, 3, 4 and 5 people so that everyone recieves the equal amount. What minimal amount of plates must Little Red Riding Hood Use?

I think that the answer is 60, but do not know how to prove it. Maybe you will be able to help me?

  • 5
    Are you splitting pies or cookies? – 2017-02-27
  • 0
    I don't understand. She has 60 pies, and by "she split pies on the plates so as to be able to split them between 2, 3, 4 or 5 people", I understand that she cut the pies in such a way as to have a number of slices divisible by 2, 3, 4, and 5. That implies that the number of slices is divisible by 60, so she can just have 5 plates and put 12 pies worth of slices on each. – 2017-02-27
  • 1
    From what we understood : she want to distribute the cookies on the minimum number of plates so that she can give plates to people and they have the same number of cookies overall. So if $2$ people, sum of plates will be $2\times 30$. If $3$ people, sum of plates will be $3\times 20$, then $4\times 15$ and finally $5\times 12$. All the different sums should be possible only by adding whole plates, it is forbidden to re-split a plate. – 2017-02-27

3 Answers 3

3

Hint: Little Red Riding Hood can definitely do better than $60$ plates, since no matter how many guests show up, at least $2$ will have at least $12$ cookies each on their plates. So she can prepare $2$ plates with $12$ cookies each and reduce the number of plates to no more than $2+(60-24)=38$.

This is equivalent to preparing $5$ plates of $12$ cookies each, and then "breaking" up $3$ of those plates into $36$ single-cookie plates. But you can "break" them up into fewer plates, that can be still combined to form $4$ $15-$cookie servings and $3$ $20-$cookie servings (note that the solution for $4$ guests automatically yields that for $2$). For example, $12,12,8,8,4,4,3,3,3,3$.

Can you do better than this solution with $10$ plates, or prove that you can't? Note that you really always need at least $8$ plates, since $4$ guests will necessarily have at least $2$ plates each (or one of the plates would hold too many cookies if $5$ guests appear).

3

It is possible to prepare 9 plates:

$12, 10, 9, 8, 7, 5, 4, 3, 2$

  • Five guests: $12, 10+2, 9+3, 8+4, 7+5$
  • Four guests: $12+3, 10+5, 9+4+2, 8+7$
  • Three guests: $12+8, 10+7+3, 9 +5+4+2$

Clearly, less than $8$ plates will not work because if one of the four guest has a single plate of 15 cookies that will be too much for one of the five guests. The remaining question is if $8$ plates will work:

The plates have to be $a, 15-a, b, 15-b, c, 15-c, d, 15-d$ without loss of generality: $3\le a \le b \le c \le d \le 7$. Clearly, two of the five guests have to have a single plate, so $a=b=3$. Since every plate has to part of $12$ and plates contain at least $3$ cookies, every plate that is not $12$ has to be at most $9$, so the possible numbers are $3,6,7$. However, each of the three guests has to have at least one plate that is not divisible by three, so $c$ and $d$ cannot be divisible by 3 and have to be 7.

The resulting $3,12,3,12,7,8,7,8$ does not work though because there is no way to fill up the 8 cookies up to 12.

Therefore, 9 plates is the best she can do.

  • 1
    I'd add, for the sake of completeness, that if there are two guests we can just combine the "four guests" parts pairwise. For example, we can have $12 + 10 + 5 + 3, 9 + 8 + 7 + 4 + 2$. – 2017-02-27
  • 0
    @Phira this is a really impressive solution. – 2017-02-27
2

I have found some other solutions (by tries and errors) with $10$ plates according to Anonymous interpretation of the problem :

Plates : $12,12,12,6,5,3,3,3,2,2$

$\begin{array}{lllll} 5\times 12 & (12) & (12) & (12) & (6,3,3) &(5,3,2,2) \\ 4\times 15 & (12,3) & (12,3) & (12,3) & (6,5,2,2) \\ 3\times 20 & (12,6,2) & (12,3,3,2) & (12,3,5) \\ 2\times 30 & (12,12,6) & (12,5,3,3,3,2,2) \\ \end{array}$

Plates : $1,2,3,4,4,4,9,10,11,12$

$\begin{array}{lllll} 5\times 12 & (12) & (11,1) & (10,2) & (9,3) &(4,4,4) \\ 4\times 15 & (12,3) & (11,4) & (10,4,1) & (9,4,2) \\ 3\times 20 & (12,4,4) & (11,9) & (10,4,3,2,1) \\ 2\times 30 & (11,10,9) & (12,4,4,4,3,2,1) \\ \end{array}$

Finally here are all the solutions for $n\le10$.

I confirm there were none for $n\le8$.

My favorites :

  • all numbers different $n=9\quad$ $1,2,4,5,7,8,10,11,12$
  • most homogeneous $n=10\quad$ $4,4,5,5,6,6,7,7,8,8$


n<=8

No solution

n=9

T = 0,1,2,3,3,7,8,12,12,12
T = 0,1,2,3,4,6,9,11,12,12
T = 0,1,2,3,4,7,8,11,12,12
T = 0,1,2,3,5,6,9,10,12,12
T = 0,1,2,3,5,7,8,10,12,12
T = 0,1,2,4,5,7,8,10,11,12
T = 0,1,3,3,4,5,9,11,12,12
T = 0,1,3,3,4,6,8,11,12,12
T = 0,1,3,3,5,7,8,9,12,12
T = 0,1,3,4,5,7,8,9,11,12
T = 0,1,3,4,6,6,8,9,11,12
T = 0,2,3,3,4,7,8,9,12,12
T = 0,2,3,3,5,6,7,10,12,12
T = 0,2,3,4,5,7,8,9,10,12
T = 0,2,3,5,6,6,7,9,10,12
T = 0,3,3,4,5,6,7,8,12,12
T = 0,3,4,5,6,6,7,8,9,12

n=10

T = 1,1,1,3,3,7,8,12,12,12
T = 1,1,1,3,4,6,9,11,12,12
T = 1,1,1,3,4,7,8,11,12,12
T = 1,1,1,3,5,6,9,10,12,12
T = 1,1,1,3,5,7,8,10,12,12
T = 1,1,1,4,4,7,8,11,11,12
T = 1,1,1,4,5,7,8,10,11,12
T = 1,1,2,2,3,7,8,12,12,12
T = 1,1,2,2,4,6,9,11,12,12
T = 1,1,2,2,4,7,8,11,12,12
T = 1,1,2,2,5,5,10,10,12,12
T = 1,1,2,2,5,6,9,10,12,12
T = 1,1,2,2,5,7,8,10,12,12
T = 1,1,2,2,6,6,9,9,12,12
T = 1,1,2,3,3,6,8,12,12,12
T = 1,1,2,3,3,6,9,11,12,12
T = 1,1,2,3,3,7,7,12,12,12
T = 1,1,2,3,3,7,8,11,12,12
T = 1,1,2,3,4,5,9,11,12,12
T = 1,1,2,3,4,5,10,10,12,12
T = 1,1,2,3,4,6,8,11,12,12
T = 1,1,2,3,4,6,9,10,12,12
T = 1,1,2,3,4,6,9,11,11,12
T = 1,1,2,3,4,7,7,11,12,12
T = 1,1,2,3,4,7,8,10,12,12
T = 1,1,2,3,4,7,8,11,11,12
T = 1,1,2,3,5,5,9,10,12,12
T = 1,1,2,3,5,6,8,10,12,12
T = 1,1,2,3,5,6,9,9,12,12
T = 1,1,2,3,5,6,9,10,11,12
T = 1,1,2,3,5,7,7,10,12,12
T = 1,1,2,3,5,7,8,9,12,12
T = 1,1,2,3,5,7,8,10,11,12
T = 1,1,2,3,6,6,8,9,12,12
T = 1,1,2,3,6,6,9,9,11,12
T = 1,1,2,4,4,6,8,11,11,12
T = 1,1,2,4,4,7,8,10,11,12
T = 1,1,2,4,5,6,8,10,11,12
T = 1,1,2,4,5,7,7,10,11,12
T = 1,1,2,4,5,7,8,9,11,12
T = 1,1,2,4,5,7,8,10,10,12
T = 1,1,2,4,5,7,8,10,11,11
T = 1,1,2,4,6,6,8,9,11,12
T = 1,1,2,5,5,7,7,10,10,12
T = 1,1,3,3,3,5,8,12,12,12
T = 1,1,3,3,3,5,9,11,12,12
T = 1,1,3,3,3,6,8,11,12,12
T = 1,1,3,3,3,6,9,11,11,12
T = 1,1,3,3,4,4,9,11,12,12
T = 1,1,3,3,4,5,8,11,12,12
T = 1,1,3,3,4,5,9,10,12,12
T = 1,1,3,3,4,5,9,11,11,12
T = 1,1,3,3,4,6,7,11,12,12
T = 1,1,3,3,4,6,8,10,12,12
T = 1,1,3,3,4,6,8,11,11,12
T = 1,1,3,3,4,7,8,9,12,12
T = 1,1,3,3,5,5,9,9,12,12
T = 1,1,3,3,5,6,7,10,12,12
T = 1,1,3,3,5,6,8,9,12,12
T = 1,1,3,3,5,6,9,9,11,12
T = 1,1,3,3,5,7,7,9,12,12
T = 1,1,3,3,5,7,8,8,12,12
T = 1,1,3,3,5,7,8,9,11,12
T = 1,1,3,3,6,6,8,8,12,12
T = 1,1,3,3,6,6,8,9,11,12
T = 1,1,3,3,6,6,9,9,11,11
T = 1,1,3,4,4,4,9,11,11,12
T = 1,1,3,4,4,5,8,11,11,12
T = 1,1,3,4,4,7,8,9,11,12
T = 1,1,3,4,5,6,8,9,11,12
T = 1,1,3,4,5,7,7,9,11,12
T = 1,1,3,4,5,7,8,8,11,12
T = 1,1,3,4,5,7,8,9,10,12
T = 1,1,3,4,5,7,8,9,11,11
T = 1,1,3,4,6,6,7,9,11,12
T = 1,1,3,4,6,6,8,8,11,12
T = 1,1,3,4,6,6,8,9,10,12
T = 1,1,3,4,6,6,8,9,11,11
T = 1,1,3,5,5,7,7,9,10,12
T = 1,1,3,5,6,6,7,9,10,12
T = 1,1,4,4,5,7,8,8,11,11
T = 1,1,4,4,6,6,8,8,11,11
T = 1,2,2,2,3,6,9,11,12,12
T = 1,2,2,2,3,7,8,11,12,12
T = 1,2,2,2,5,5,10,10,11,12
T = 1,2,2,2,5,7,8,10,11,12
T = 1,2,2,3,3,5,8,12,12,12
T = 1,2,2,3,3,5,9,11,12,12
T = 1,2,2,3,3,5,10,10,12,12
T = 1,2,2,3,3,6,7,12,12,12
T = 1,2,2,3,3,6,8,11,12,12
T = 1,2,2,3,3,6,9,10,12,12
T = 1,2,2,3,3,7,7,11,12,12
T = 1,2,2,3,3,7,8,10,12,12
T = 1,2,2,3,4,4,9,11,12,12
T = 1,2,2,3,4,5,8,11,12,12
T = 1,2,2,3,4,5,9,10,12,12
T = 1,2,2,3,4,5,10,10,11,12
T = 1,2,2,3,4,6,7,11,12,12
T = 1,2,2,3,4,6,9,9,12,12
T = 1,2,2,3,4,6,9,10,11,12
T = 1,2,2,3,4,7,8,9,12,12
T = 1,2,2,3,4,7,8,10,11,12
T = 1,2,2,3,5,5,8,10,12,12
T = 1,2,2,3,5,5,9,10,11,12
T = 1,2,2,3,5,6,7,10,12,12
T = 1,2,2,3,5,6,8,9,12,12
T = 1,2,2,3,5,6,9,10,10,12
T = 1,2,2,3,5,7,7,10,11,12
T = 1,2,2,3,5,7,8,8,12,12
T = 1,2,2,3,5,7,8,9,11,12
T = 1,2,2,3,5,7,8,10,10,12
T = 1,2,2,3,6,6,7,9,12,12
T = 1,2,2,3,6,6,8,9,11,12
T = 1,2,2,3,6,6,9,9,10,12
T = 1,2,2,4,5,5,8,10,11,12
T = 1,2,2,4,5,6,7,10,11,12
T = 1,2,2,4,5,7,8,8,11,12
T = 1,2,2,4,5,7,8,9,10,12
T = 1,2,2,4,5,7,8,10,10,11
T = 1,2,2,5,5,6,7,10,10,12
T = 1,2,2,5,5,7,7,10,10,11
T = 1,2,2,5,6,6,7,9,10,12
T = 1,2,3,3,3,4,8,12,12,12
T = 1,2,3,3,3,4,9,11,12,12
T = 1,2,3,3,3,5,7,12,12,12
T = 1,2,3,3,3,5,9,10,12,12
T = 1,2,3,3,3,6,7,11,12,12
T = 1,2,3,3,3,6,8,10,12,12
T = 1,2,3,3,3,6,9,10,11,12
T = 1,2,3,3,3,7,8,9,12,12
T = 1,2,3,3,4,4,7,12,12,12
T = 1,2,3,3,4,4,8,11,12,12
T = 1,2,3,3,4,4,9,10,12,12
T = 1,2,3,3,4,5,6,12,12,12
T = 1,2,3,3,4,5,7,11,12,12
T = 1,2,3,3,4,5,8,10,12,12
T = 1,2,3,3,4,5,9,9,12,12
T = 1,2,3,3,4,5,9,10,11,12
T = 1,2,3,3,4,6,6,11,12,12
T = 1,2,3,3,4,6,7,10,12,12
T = 1,2,3,3,4,6,8,9,12,12
T = 1,2,3,3,4,6,8,10,11,12
T = 1,2,3,3,4,6,9,9,11,12
T = 1,2,3,3,4,7,7,9,12,12
T = 1,2,3,3,4,7,8,8,12,12
T = 1,2,3,3,4,7,8,9,11,12
T = 1,2,3,3,5,5,5,12,12,12
T = 1,2,3,3,5,5,7,10,12,12
T = 1,2,3,3,5,5,8,9,12,12
T = 1,2,3,3,5,6,6,10,12,12
T = 1,2,3,3,5,6,7,9,12,12
T = 1,2,3,3,5,6,7,10,11,12
T = 1,2,3,3,5,6,9,9,10,12
T = 1,2,3,3,5,7,7,8,12,12
T = 1,2,3,3,5,7,8,9,10,12
T = 1,2,3,3,6,6,7,8,12,12
T = 1,2,3,3,6,6,7,9,11,12
T = 1,2,3,3,6,6,8,9,10,12
T = 1,2,3,3,6,6,9,9,10,11
T = 1,2,3,4,4,4,7,11,12,12
T = 1,2,3,4,4,4,9,10,11,12
T = 1,2,3,4,4,5,6,11,12,12
T = 1,2,3,4,4,5,7,10,12,12
T = 1,2,3,4,4,5,8,10,11,12
T = 1,2,3,4,4,6,7,9,12,12
T = 1,2,3,4,4,6,8,9,11,12
T = 1,2,3,4,4,7,7,8,12,12
T = 1,2,3,4,4,7,8,8,11,12
T = 1,2,3,4,4,7,8,9,10,12
T = 1,2,3,4,5,5,5,11,12,12
T = 1,2,3,4,5,5,6,10,12,12
T = 1,2,3,4,5,5,7,10,11,12
T = 1,2,3,4,5,5,8,9,11,12
T = 1,2,3,4,5,6,6,9,12,12
T = 1,2,3,4,5,6,6,10,11,12
T = 1,2,3,4,5,6,7,8,12,12
T = 1,2,3,4,5,6,7,9,11,12
T = 1,2,3,4,5,6,8,9,10,12
T = 1,2,3,4,5,7,7,8,11,12
T = 1,2,3,4,5,7,7,9,10,12
T = 1,2,3,4,5,7,8,8,10,12
T = 1,2,3,4,5,7,8,9,9,12
T = 1,2,3,4,5,7,8,9,10,11
T = 1,2,3,4,6,6,6,9,11,12
T = 1,2,3,4,6,6,7,8,11,12
T = 1,2,3,4,6,6,7,9,10,12
T = 1,2,3,4,6,6,8,9,9,12
T = 1,2,3,4,6,6,8,9,10,11
T = 1,2,3,5,5,5,6,9,12,12
T = 1,2,3,5,5,5,7,8,12,12
T = 1,2,3,5,5,6,7,9,10,12
T = 1,2,3,5,5,7,7,8,10,12
T = 1,2,3,5,6,6,6,9,10,12
T = 1,2,3,5,6,6,7,8,10,12
T = 1,2,3,5,6,6,7,9,9,12
T = 1,2,3,5,6,6,7,9,10,11
T = 1,2,4,4,4,5,7,10,11,12
T = 1,2,4,4,5,6,7,8,11,12
T = 1,2,4,4,5,7,7,8,10,12
T = 1,2,4,4,5,7,8,8,10,11
T = 1,2,4,5,5,5,7,8,11,12
T = 1,2,4,5,5,6,7,8,10,12
T = 1,2,4,5,5,7,7,8,10,11
T = 1,2,4,5,6,6,7,8,9,12
T = 1,2,4,5,6,6,7,8,10,11
T = 1,3,3,3,3,4,8,11,12,12
T = 1,3,3,3,3,5,7,11,12,12
T = 1,3,3,3,4,5,5,12,12,12
T = 1,3,3,3,4,5,6,11,12,12
T = 1,3,3,3,4,5,8,9,12,12
T = 1,3,3,3,4,5,9,9,11,12
T = 1,3,3,3,4,6,8,8,12,12
T = 1,3,3,3,4,6,8,9,11,12
T = 1,3,3,3,5,5,7,9,12,12
T = 1,3,3,3,5,6,7,8,12,12
T = 1,3,3,3,5,6,7,9,11,12
T = 1,3,3,3,5,7,8,9,9,12
T = 1,3,3,4,4,4,6,11,12,12
T = 1,3,3,4,4,5,5,11,12,12
T = 1,3,3,4,4,5,7,9,12,12
T = 1,3,3,4,4,5,8,9,11,12
T = 1,3,3,4,4,6,7,8,12,12
T = 1,3,3,4,4,6,8,8,11,12
T = 1,3,3,4,5,5,6,9,12,12
T = 1,3,3,4,5,5,7,8,12,12
T = 1,3,3,4,5,5,7,9,11,12
T = 1,3,3,4,5,6,6,8,12,12
T = 1,3,3,4,5,6,6,9,11,12
T = 1,3,3,4,5,6,7,7,12,12
T = 1,3,3,4,5,6,7,8,11,12
T = 1,3,3,4,5,6,8,9,9,12
T = 1,3,3,4,5,7,8,8,9,12
T = 1,3,3,4,5,7,8,9,9,11
T = 1,3,3,4,6,6,6,8,11,12
T = 1,3,3,4,6,6,8,8,9,12
T = 1,3,3,4,6,6,8,9,9,11
T = 1,3,3,5,5,5,7,7,12,12
T = 1,3,3,5,5,6,7,9,9,12
T = 1,3,3,5,5,7,7,8,9,12
T = 1,3,3,5,6,6,7,8,9,12
T = 1,3,3,5,6,6,7,9,9,11
T = 1,3,4,4,4,5,7,9,11,12
T = 1,3,4,4,4,6,6,9,11,12
T = 1,3,4,4,5,5,7,8,11,12
T = 1,3,4,4,5,6,6,8,11,12
T = 1,3,4,4,5,7,7,8,9,12
T = 1,3,4,4,5,7,8,8,9,11
T = 1,3,4,4,6,6,7,8,9,12
T = 1,3,4,4,6,6,8,8,9,11
T = 1,3,4,5,5,5,7,7,11,12
T = 1,3,4,5,5,6,7,8,9,12
T = 1,3,4,5,5,7,7,8,8,12
T = 1,3,4,5,5,7,7,8,9,11
T = 1,3,4,5,6,6,6,8,9,12
T = 1,3,4,5,6,6,7,7,9,12
T = 1,3,4,5,6,6,7,8,8,12
T = 1,3,4,5,6,6,7,8,9,11
T = 1,3,4,6,6,6,6,8,9,11
T = 1,3,5,5,5,6,7,7,9,12
T = 1,3,5,5,5,7,7,7,8,12
T = 1,4,4,5,5,7,7,8,8,11
T = 1,4,4,5,6,6,7,8,8,11
T = 1,4,5,5,5,7,7,7,8,11
T = 2,2,2,3,3,6,9,9,12,12
T = 2,2,2,3,3,7,8,9,12,12
T = 2,2,2,3,5,5,9,10,10,12
T = 2,2,2,3,5,7,8,9,10,12
T = 2,2,3,3,3,3,8,12,12,12
T = 2,2,3,3,3,3,10,10,12,12
T = 2,2,3,3,3,5,6,12,12,12
T = 2,2,3,3,3,6,7,10,12,12
T = 2,2,3,3,3,6,8,9,12,12
T = 2,2,3,3,3,6,9,10,10,12
T = 2,2,3,3,4,4,9,9,12,12
T = 2,2,3,3,4,5,7,10,12,12
T = 2,2,3,3,4,5,8,9,12,12
T = 2,2,3,3,4,5,9,10,10,12
T = 2,2,3,3,4,6,7,9,12,12
T = 2,2,3,3,4,6,9,9,10,12
T = 2,2,3,3,4,7,7,8,12,12
T = 2,2,3,3,4,7,8,9,10,12
T = 2,2,3,3,5,5,6,10,12,12
T = 2,2,3,3,5,6,6,9,12,12
T = 2,2,3,3,5,6,7,8,12,12
T = 2,2,3,3,5,6,7,10,10,12
T = 2,2,3,3,6,6,7,7,12,12
T = 2,2,3,3,6,6,7,9,10,12
T = 2,2,3,3,6,6,8,9,9,12
T = 2,2,3,3,6,6,9,9,10,10
T = 2,2,3,4,5,5,7,10,10,12
T = 2,2,3,4,5,5,8,9,10,12
T = 2,2,3,4,5,6,7,9,10,12
T = 2,2,3,4,5,7,7,8,10,12
T = 2,2,3,4,5,7,8,8,9,12
T = 2,2,3,4,5,7,8,9,10,10
T = 2,2,3,5,5,6,6,9,10,12
T = 2,2,3,5,6,6,7,7,10,12
T = 2,2,3,5,6,6,7,8,9,12
T = 2,2,3,5,6,6,7,9,10,10
T = 2,2,4,5,5,7,7,8,10,10
T = 2,2,5,5,6,6,7,7,10,10
T = 2,3,3,3,3,4,8,10,12,12
T = 2,3,3,3,3,5,5,12,12,12
T = 2,3,3,3,3,5,7,10,12,12
T = 2,3,3,3,4,4,5,12,12,12
T = 2,3,3,3,4,4,8,9,12,12
T = 2,3,3,3,4,5,6,10,12,12
T = 2,3,3,3,4,5,7,9,12,12
T = 2,3,3,3,4,5,9,9,10,12
T = 2,3,3,3,4,6,7,8,12,12
T = 2,3,3,3,4,6,8,9,10,12
T = 2,3,3,3,4,7,8,9,9,12
T = 2,3,3,3,5,5,6,9,12,12
T = 2,3,3,3,5,6,7,7,12,12
T = 2,3,3,3,5,6,7,9,10,12
T = 2,3,3,4,4,4,7,9,12,12
T = 2,3,3,4,4,5,5,10,12,12
T = 2,3,3,4,4,5,6,9,12,12
T = 2,3,3,4,4,5,7,8,12,12
T = 2,3,3,4,4,5,8,9,10,12
T = 2,3,3,4,4,6,8,9,9,12
T = 2,3,3,4,4,7,8,8,9,12
T = 2,3,3,4,5,5,5,9,12,12
T = 2,3,3,4,5,5,6,8,12,12
T = 2,3,3,4,5,5,7,9,10,12
T = 2,3,3,4,5,6,6,7,12,12
T = 2,3,3,4,5,6,6,9,10,12
T = 2,3,3,4,5,6,7,8,10,12
T = 2,3,3,4,5,6,7,9,9,12
T = 2,3,3,4,5,7,7,8,9,12
T = 2,3,3,4,5,7,8,9,9,10
T = 2,3,3,4,6,6,7,8,9,12
T = 2,3,3,4,6,6,8,9,9,10
T = 2,3,3,5,5,5,6,7,12,12
T = 2,3,3,5,5,6,6,9,9,12
T = 2,3,3,5,5,6,7,7,10,12
T = 2,3,3,5,6,6,6,7,10,12
T = 2,3,3,5,6,6,7,7,9,12
T = 2,3,3,5,6,6,7,9,9,10
T = 2,3,4,4,4,5,7,9,10,12
T = 2,3,4,4,5,5,7,8,10,12
T = 2,3,4,4,5,6,7,8,9,12
T = 2,3,4,4,5,7,7,8,8,12
T = 2,3,4,4,5,7,8,8,9,10
T = 2,3,4,5,5,5,7,8,9,12
T = 2,3,4,5,5,6,6,7,10,12
T = 2,3,4,5,5,6,6,8,9,12
T = 2,3,4,5,5,7,7,8,9,10
T = 2,3,4,5,6,6,6,7,9,12
T = 2,3,4,5,6,6,7,7,8,12
T = 2,3,4,5,6,6,7,8,9,10
T = 2,3,5,5,5,6,6,7,9,12
T = 2,3,5,5,6,6,7,7,9,10
T = 2,3,5,6,6,6,6,7,9,10
T = 2,4,4,5,5,7,7,8,8,10
T = 2,4,5,5,6,6,7,7,8,10
T = 3,3,3,3,4,4,8,8,12,12
T = 3,3,3,3,4,5,7,8,12,12
T = 3,3,3,3,5,5,7,7,12,12
T = 3,3,3,4,4,5,6,8,12,12
T = 3,3,3,4,4,6,8,8,9,12
T = 3,3,3,4,5,5,6,7,12,12
T = 3,3,3,4,5,6,7,8,9,12
T = 3,3,3,5,5,6,7,7,9,12
T = 3,3,4,4,4,5,6,7,12,12
T = 3,3,4,4,5,5,5,7,12,12
T = 3,3,4,4,5,5,6,6,12,12
T = 3,3,4,4,5,6,6,8,9,12
T = 3,3,4,4,5,6,7,8,8,12
T = 3,3,4,4,6,6,8,8,9,9
T = 3,3,4,5,5,6,6,7,9,12
T = 3,3,4,5,5,6,7,7,8,12
T = 3,3,4,5,6,6,6,7,8,12
T = 3,3,4,5,6,6,7,8,9,9
T = 3,3,5,5,6,6,7,7,9,9
T = 3,4,4,4,5,6,6,7,9,12
T = 3,4,4,5,5,5,7,7,8,12
T = 3,4,4,5,5,6,6,7,8,12
T = 3,4,4,5,6,6,7,8,8,9
T = 3,4,5,5,6,6,7,7,8,9
T = 3,4,5,6,6,6,6,7,8,9
T = 4,4,5,5,5,7,7,7,8,8
T = 4,4,5,5,6,6,7,7,8,8