Ten objects o1, o2, …, o10 were distributed among Amar, Barat, Charles, Disha, and Elise. Each item went to exactly one person. Each person got exactly two of the items, and this pair of objects is called her/his bundle.
The following table shows how each person values each object.
The value of any bundle by a person is the sum of that person's values of the objects in that bundle. A person X envies another person Y if X values Y 's bundle more than X 's own bundle.
For example, hypothetically suppose Amar's bundle consists of and , and Barat's bundle consists of and . Then Amar values his own bundle at and Barat's bundle at
. Hence Amar does not envy Barat. On the other hand, Barat values his own bundle at and Amar's bundle at . Hence Barat envies Amar.
The following facts are known about the actual distribution of the objects among the five people.
- If someone's value for an object is , then she/he received that object.
- Objects o1, o2, and o3 were given to three different people.
- Objects and were given to different people.
- Three people value their own bundles at . No one values her/his own bundle at a number higher than .
- Disha values her own bundle at an odd number. All others value their own bundles at an even number.
- Some people who value their own bundles less than envy some other people who value their own bundle at . No one else envies others.
Solution
| o1 | o2 | o3 | o4 | o5 | o6 | o7 | o8 | o9 | o10 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Amar | 4 | 9 | 9 | 3 | 7 | 3 | 8 | 7 | 9 | 5 |
| Barat | 5 | 9 | 7 | 5 | 5 | 3 | 6 | 8 | 10 | 8 |
| Charles | 8 | 8 | 8 | 3 | 6 | 4 | 5 | 8 | 9 | 6 |
| Disha | 8 | 8 | 8 | 5 | 5 | 3 | 6 | 4 | 9 | 8 |
| Elise | 6 | 8 | 9 | 5 | 6 | 5 | 6 | 3 | 7 | 10 |
Understanding the set :-
- There are 10 objects to be distributed among Amar, Barat, Charles, Disha, and Elise.
- Each person got exactly two of the items, and this pair of objects is called her/his bundle.
- This table is given in the question, which gives a number showing how much a person values that product.
(for eg - o1 - valued by Amar as 4, while, the value of the same object, o1 for Disha is 8)
- value of any bundle by a person is the sum of that person's values of the objects in that bundle
- A person X envies another person Y if X values Y's bundle more than X's own bundle.
(For eg - if Amar's bundle be o3 and o4 = 9 + 3 = 12 and Disha values Amar bundle as o3 + o4 = 8 + 5 = 13
=> Disha values Amar's bundle more then Amar
=> Dsha will envy Amar)
In this set, we need to find the bundle of all the 5 people, and also, if anyone envy other.
| o1 | o2 | o3 | o4 | o5 | o6 | o7 | o8 | o9 | o10 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Amar | 4 | 9 | 9 | 3 | 7 | 3 | 8 | 7 | 9 | 5 |
| Barat | 5 | 9 | 7 | 5 | 5 | 3 | 6 | 8 | 10 | 8 |
| Charles | 8 | 8 | 8 | 3 | 6 | 4 | 5 | 8 | 9 | 6 |
| Disha | 8 | 8 | 8 | 5 | 5 | 3 | 6 | 4 | 9 | 8 |
| Elise | 6 | 8 | 9 | 5 | 6 | 5 | 6 | 3 | 7 | 10 |
| Object | Object | Total | |
|---|---|---|---|
| Amar | |||
| Bharat | o9(10) | ||
| Charles | |||
| Disha | |||
| Elise | o10 (10) |
Clue 1 says,
If someone's value for an object is 10, then she/he received that object.
=> Bharat got object 9
and Elise got object 10
| o1 | o2 | o3 | o4 | o5 | o6 | o7 | o8 | o9 | o10 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Amar | 4 | 9 | 9 | 3 | 7 | 3 | 8 | 7 | 9 | 5 |
| Barat | 5 | 9 | 7 | 5 | 5 | 3 | 6 | 8 | 10 | 8 |
| Charles | 8 | 8 | 8 | 3 | 6 | 4 | 5 | 8 | 9 | 6 |
| Disha | 8 | 8 | 8 | 5 | 5 | 3 | 6 | 4 | 9 | 8 |
| Elise | 6 | 8 | 9 | 5 | 6 | 5 | 6 | 3 | 7 | 10 |
| Object | Object | Total | |
|---|---|---|---|
| Amar | |||
| Bharat | o9 (10) | ||
| Charles | |||
| Disha | |||
| Elise | o10 (10) |
Now, in the table we don't need to look at o9 and o10 column as we already had found to whom they belong.
clue 4 says,
Three people value their own bundles at 16. No one values her/his own bundle at a number higher than 16.
Also, clue 5 says,
Disha values her own bundle at an odd number. All others value their own bundles at an even number.
=>There are three people valuing their bundles - 16, and four of them value their bundle at an even number except Disha.
=> Elise and Bharat also values their objects at an odd number
=> Possibilities for another object for Bharat - o7 or o8
and, Possibilities for another object for Eliza - o1 or o2 or o5 or o7
| o1 | o2 | o3 | o4 | o5 | o6 | o7 | o8 | o9 | o10 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Amar | 4 | 9 | 9 | 3 | 7 | 3 | 8 | 7 | 9 | 5 |
| Barat | 5 | 9 | 7 | 5 | 5 | 3 | 6 | 8 | 10 | 8 |
| Charles | 8 | 8 | 8 | 3 | 6 | 4 | 5 | 8 | 9 | 6 |
| Disha | 8 | 8 | 8 | 5 | 5 | 3 | 6 | 4 | 9 | 8 |
| Elise | 6 | 8 | 9 | 5 | 6 | 5 | 6 | 3 | 7 | 10 |
| Object | Object | Total | |
|---|---|---|---|
| Amar | |||
| Bharat | o9 (10) | o7 (6) | 16 |
| Charles | |||
| Disha | |||
| Elise | o10 (10) |
Continuing with clue 4,
Three people value their own bundles at 16. No one values her/his own bundle at a number higher than 16.
For Bharat :
Possibility = o7 or o8
But if it is o8 then total will become 10 (o9) + 8 (o8) will be greater then 16, which is not possible
Therefore, Bharat had o9 and o7 .
Similarly for Elisa,
We know Elisa can have o1, o2 or o5 (o7 is now with Bharat already)
But it can't be o2 as it will make the sum greater then 16
=> Other object with Elisa will be o1 or o5
But Elisa values both of them 6
Therefore, total of values of Elisa = 16
| o1 | o2 | o3 | o4 | o5 | o6 | o7 | o8 | o9 | o10 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Amar | 4 | 9 | 9 | 3 | 7 | 3 | 8 | 7 | 9 | 5 |
| Barat | 5 | 9 | 7 | 5 | 5 | 3 | 6 | 8 | 10 | 8 |
| Charles | 8 | 8 | 8 | 3 | 6 | 4 | 5 | 8 | 9 | 6 |
| Disha | 8 | 8 | 8 | 5 | 5 | 3 | 6 | 4 | 9 | 8 |
| Elise | 6 | 8 | 9 | 5 | 6 | 5 | 6 | 3 | 7 | 10 |
| Object | Object | Total | |
|---|---|---|---|
| Amar | |||
| Bharat | o9 (10) | o7 (6) | 16 |
| Charles | o8 (8) | o2 / o3 (8) | 16 |
| Disha | |||
| Elise | o10 (10) | o1 / o5 (6) | 16 |
Clue 6 says,
Some people who value their own bundles less than 16 envy some other people who value their own bundle at 16. No one else envies others.
For Amar :-
Amar values Barat’s bundle at 17. (9 + 8)
So, Amar could not have scored 16, otherwise he would have envied Barat.
=> Charles scores 16.
Charles, Barat and Elise score 16 each.
Now, Charles 16 should come from two 8’s.
O1 cannot comes with o2, o3 or o8.
O2 and O3 cannot come together.
So, o8 should be accompanied by o2 or o3.
| o1 | o2 | o3 | o4 | o5 | o6 | o7 | o8 | o9 | o10 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Amar | 4 | 9 | 9 | 3 | 7 | 3 | 8 | 7 | 9 | 5 |
| Barat | 5 | 9 | 7 | 5 | 5 | 3 | 6 | 8 | 10 | 8 |
| Charles | 8 | 8 | 8 | 3 | 6 | 4 | 5 | 8 | 9 | 6 |
| Disha | 8 | 8 | 8 | 5 | 5 | 3 | 6 | 4 | 9 | 8 |
| Elise | 6 | 8 | 9 | 5 | 6 | 5 | 6 | 3 | 7 | 10 |
| Object | Object | Total | |
|---|---|---|---|
| Amar | |||
| Bharat | o9 (10) | o7 (6) | 16 |
| Charles | o8 (8) | o3 (8) | 16 |
| Disha | |||
| Elise | o10 (10) | o1 / o5 (6) | 16 |
Now, If Charles had got o8 and o2,
Barat would have valued Charles bundle at 17 and envied it.
This cannot happen.
So, Charles should have gotten O8 and O3.
| o1 | o2 | o3 | o4 | o5 | o6 | o7 | o8 | o9 | o10 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Amar | 4 | 9 | 9 | 3 | 7 | 3 | 8 | 7 | 9 | 5 |
| Barat | 5 | 9 | 7 | 5 | 5 | 3 | 6 | 8 | 10 | 8 |
| Charles | 8 | 8 | 8 | 3 | 6 | 4 | 5 | 8 | 9 | 6 |
| Disha | 8 | 8 | 8 | 5 | 5 | 3 | 6 | 4 | 9 | 8 |
| Elise | 6 | 8 | 9 | 5 | 6 | 5 | 6 | 3 | 7 | 10 |
| Object | Object | Total | |
|---|---|---|---|
| Amar | |||
| Bharat | o9 (10) | o7 (6) | 16 |
| Charles | o8 (8) | o3 (8) | 16 |
| Disha | |||
| Elise | o10 (10) | o1 / o5 (6) | 16 |
Now, we have o1, o2, o4, o5, o6 remaining
2 for Amar,
2 for Disha
and 1 for Elise.
Elise should have gotten o1 or o5.
Amar should have got some two of o2, o4, o5, o6.
Disha should have got one of o1, o2 and one of o4, o5, o6.
Let us break this into two possibilities - E getting o1 or E getting o5.
CASE - 1 :- E getting o1
| o1 | o2 | o3 | o4 | o5 | o6 | o7 | o8 | o9 | o10 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Amar | 4 | 9 | 9 | 3 | 7 | 3 | 8 | 7 | 9 | 5 |
| Barat | 5 | 9 | 7 | 5 | 5 | 3 | 6 | 8 | 10 | 8 |
| Charles | 8 | 8 | 8 | 3 | 6 | 4 | 5 | 8 | 9 | 6 |
| Disha | 8 | 8 | 8 | 5 | 5 | 3 | 6 | 4 | 9 | 8 |
| Elise | 6 | 8 | 9 | 5 | 6 | 5 | 6 | 3 | 7 | 10 |
| Object | Object | Total | |
|---|---|---|---|
| Amar | |||
| Bharat | o9 (10) | o7 (6) | 16 |
| Charles | o8 (8) | o3 (8) | 16 |
| Disha | o2 (8) | ||
| Elise | o10 (10) | o1 (6) | 16 |
Now according to clue 2,
Objects o1, o2, and o3 were given to three different people.
Therefore, o2 will go to Disha
Then, Amar has to get 3 + 7, or 3 + 3.
Amar values Disha’s o2 at 9.
So, Amar will end up valuing Disha’s bundle at higher than this own.
Or, Amar will envy Disha.
=> This is not possible.
Therefore, Elisa will have o5.
| o1 | o2 | o3 | o4 | o5 | o6 | o7 | o8 | o9 | o10 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Amar | 4 | 9 | 9 | 3 | 7 | 3 | 8 | 7 | 9 | 5 |
| Barat | 5 | 9 | 7 | 5 | 5 | 3 | 6 | 8 | 10 | 8 |
| Charles | 8 | 8 | 8 | 3 | 6 | 4 | 5 | 8 | 9 | 6 |
| Disha | 8 | 8 | 8 | 5 | 5 | 3 | 6 | 4 | 9 | 8 |
| Elise | 6 | 8 | 9 | 5 | 6 | 5 | 6 | 3 | 7 | 10 |
| Object | Object | Total | |
|---|---|---|---|
| Amar | 02 (9) | o6 (3) | 12 |
| Bharat | o9 (10) | o7 (6) | 16 |
| Charles | o8 (8) | o3 (8) | 16 |
| Disha | o1 (8) | o4 (5) | 13 |
| Elise | o10 (10) | o5 (6) | 16 |
Now, since Elisa have o5
=> Disha will have o1
and, Amar will have 02
Now Amar has to get one of 04 or 06 and Disha has to get the other.
If Amar gets o4, his score would be 12.
Disha’s score would be 11 and Disha would value Amar’s bundle at 13
(8 for o2 and 5 for o4).
This would make Disha envious of Amar, which is not possible.
So, Amar should get o6 and Disha 04.
| Object | Object | Total | |
|---|---|---|---|
| Amar | 02 (9) | o6 (3) | 12 |
| Bharat | o9 (10) | o7 (6) | 16 |
| Charles | o8 (8) | o3 (8) | 16 |
| Disha | o1 (8) | o4 (5) | 13 |
| Elise | o10 (10) | o5 (6) | 16 |
Now question can be answered using the above table.
Amar's bundle value = 12
More from this set:
Question 15
Question 16
Question 18
Question 19