A9740 | Candy Boxes

There is an old tradition of keeping $4$ boxes of candies in the house in Cyberland. The numbers of candies are special if their arithmetic mean, their median and their range are all equal. By definition, for a set ${x_{1},x_{2},x_{3},x_{4}}$ ( $x_{1}<=x_{2}<=x_{3}<=x_{4}$ ) arithmetic mean is , median is  and range is $x_{4}-x_{1}$ . The arithmetic mean and median are not necessary integer. It is well-known that if those three numbers are same, boxes will create a "debugging field" and codes in the field will have no bugs.

For example, $1,1,3,3$ is the example of $4$ numbers meeting the condition because their mean, median and range are all equal to $2$ .

Jeff has $4$ special boxes of candies. However, something bad has happened! Some of the boxes could have been lost and now there are only $n$ ( $0<=n<=4$ ) boxes remaining. The $i$ -th remaining box contains $a_{i}$ candies.

Now Jeff wants to know: is there a possible way to find the number of candies of the $4-n$ missing boxes, meeting the condition above (the mean, median and range are equal)?

There is an old tradition of keeping $4$ boxes of candies in the house in Cyberland. The numbers of candies are special if their arithmetic mean, their median and their range are all equal. By definition, for a set ${x_{1},x_{2},x_{3},x_{4}}$ ( $x_{1}<=x_{2}<=x_{3}<=x_{4}$ ) arithmetic mean is , median is  and range is $x_{4}-x_{1}$ . The arithmetic mean and median are not necessary integer. It is well-known that if those three numbers are same, boxes will create a "debugging field" and codes in the field will have no bugs.

For example, $1,1,3,3$ is the example of $4$ numbers meeting the condition because their mean, median and range are all equal to $2$ .

Jeff has $4$ special boxes of candies. However, something bad has happened! Some of the boxes could have been lost and now there are only $n$ ( $0<=n<=4$ ) boxes remaining. The $i$ -th remaining box contains $a_{i}$ candies.

Now Jeff wants to know: is there a possible way to find the number of candies of the $4-n$ missing boxes, meeting the condition above (the mean, median and range are equal)?

There is an old tradition of keeping $4$ boxes of candies in the house in Cyberland. The numbers of candies are special if their arithmetic mean, their median and their range are all equal. By definition, for a set ${x_{1},x_{2},x_{3},x_{4}}$ ( $x_{1}<=x_{2}<=x_{3}<=x_{4}$ ) arithmetic mean is , median is  and range is $x_{4}-x_{1}$ . The arithmetic mean and median are not necessary integer. It is well-known that if those three numbers are same, boxes will create a "debugging field" and codes in the field will have no bugs.

For example, $1,1,3,3$ is the example of $4$ numbers meeting the condition because their mean, median and range are all equal to $2$ .

Jeff has $4$ special boxes of candies. However, something bad has happened! Some of the boxes could have been lost and now there are only $n$ ( $0<=n<=4$ ) boxes remaining. The $i$ -th remaining box contains $a_{i}$ candies.

Now Jeff wants to know: is there a possible way to find the number of candies of the $4-n$ missing boxes, meeting the condition above (the mean, median and range are equal)?