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Let be the number of robots designed on the th day. Then we have 30 numbers and for all .
Let , it is the number of robots designed on or before the th day.
We have a sequence of 30 numbers .
Let us consider new sequence: . There are 30 numbers and .
There are totally 60 numbers: and all of them are less or equal than ( and ).
By the Pigeonhole Principle, at least two of these numbers are equal.
Since and for all , it follows that for some and .
Now we can conclude that exactly 14 robots were designed from day to day .