There are $1024$ people. Each person is having $1024$ similar fruits. No two persons are having same kind of fruits. Find the minimum number of transactions required so that each-person is having $1024$ different kinds offruits. Note that a single transaction is defined as exchange of any number of fruits between any two persons.
The suggested solution is as follows:
If $2$ people having $2$ fruits each then only one transaction will be required. If $4$ people having $4$ fruits each then $2 \times 2$ transactions are required. $8$ people having $8$ fruits each then $3 \times 4$ transactions will be required.Like this for $1024$ we would need a minimum of $10 \times 2^9 = 5120$
But I am not sure that I can fully construe this.Lets denote the fruits by alphabets,For two people initially we have
Hence,only one exchange of 'a' with 'b',will make it
But when when we have four people then,
aaaa bbbb cccc dddd
I tried but I could not make this to
abcd abcd abcd abcd
in only $4$ steps,could anybody show me how to do this in $4$ steps?