For any transferable utility game in coalitional form with nonempty core, we show that the number of blocks required to switch from an imputation out of the core to a core imputation is less than or equal to n(n−1)/2, where n is the number of players. This number considerably improves the bounds found by Kóczy (2006) [5] and Yang (2010) [11]. Our result relies on an altered version of the procedure proposed by Sengupta and Sengupta (1996) [9]. The use of the Davis-Maschler reduced-games is also pointed out.