Unfolded Truth

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*   *       1
    *   *   
        *   *

*   *       2
    *   *   *
        *   

*   *       3
    *   *   *
    *       

*           4
*   *   *   *
    *       

    *       5
*   *   *   *
        *   

*   *       6
    *   *   *
            *

*           7
*   *   *   *
*           

*           8
*   *   *   *
        *   

*           9   
*   *   *   *   
            *   

    *       10  
*   *   *   *   
    *           

*   *   *       11
        *   *   *

It ought to be thin that joint of squares must only be one edge with others. Where no sticking edges required, 10 of them would need a (3 $\times$ 4) rectangle to cut where 1 of them would need only a (2 $\times$ 5) rectangle to cut. For longest strip of 4 straight squares, it makes a complete round and therefore (4 + 2) cannot be there but only with (4 + 1 + 1):

One (2 + 2 + 2)

Three (3 + 2 + 1)

Six (4 + 1 + 1)

One (3 + 3)

The answer should be correct. Everyone is having 5 joining edges while every cube makes a 12 edges.

Answer: $\boxed{11}$