Every day the piece is reduced by 20%. In how many days will the bar of soap be reduced by more than half?

Let’s denote the size of the original bar of soap through x.

The problem statement says that every day the bar of soap is reduced by 20%, therefore, after one day, the size of the remaining bar of soap will be x – (20/100) x = x – (2/10) x = x – 0.2x = 0.8x.

After another day, the size of the remaining bar of soap will be 0.8x – (20/100) * 0.8x = 0.8x – 0.2 * 0.8x = 0.8x – 0.16x = 0.64x.

In another day, the size of the remaining bar of soap will be 0.64x – (20/100) * 0.64x = 0.64x – 0.2 * 0.64x = 0.64x – 0.128x = 0.512x.

After another day, the size of the remaining bar of soap will be 0.512x – (20/100) * 0.512x = 0.512x – 0.2 * 0.512x = 0.512x – 0.1024x = 0.4096x.

Thus, the bar of soap will shrink by more than half after 4 days.

Answer: in 4 days.