Snowman Identity

In Little Mole in Winter, Krtek builds a Snowman, but one sunny day Snowman's body melts away. Krtek saves Snowman at a point where only Snowman's head remains by making new body balls from nearby (fresh) snow. Now imagine that Snowman melts away completely, but you are experiencing this and make an attempt to re-make Snowman from roughly the same pile of snow. Is it going to be the same Snowman as before the meltdown?