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2019 NSW: Maths Puzzle 2

2nd puzzle by Dr Harry Wiggins (Helping Hercules)

Hydra is a serpentine water monster that lives in the lake of Lerna. It is Hercules's task to kill this monster (the second task of his Twelve Labors).  The Hydra possesses 100 heads and can only be killed by cutting of ALL his heads. It is know that you can cut off exactly 10, 20, 30 or 40 heads at a time. However with each of these 40, 2, 50 or 4 new heads will re-appear in each case. 

How can Hercules defeat the monster? What is the least number of cuts Hercules needs?



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2019/07/30 3:05 PM
1) Cut off 40 heads and then you'll be left with 64
2) Cut off 40 again and then you'll be left with 28
3) Cut off 20 and then you'll be left with 10
4) Finally cut off the remaining 10 heads

The least number of cuts is 4
2019/07/30 4:11 PM
Wouldn't cutting off the last 10 heads (step 4 in Lloyd's comment ) make 40 new heads to re-appear?
2019/07/30 5:00 PM
Lloyd's solution is correct. Maybe we need to add to the problem description, that if all the heads are cut off, then no new heads will grow.

My follow up question is, how do we know that 4 cuts is the best? That is, is there not a solution using 3 cuts? Who can PROVE that 4 is the best or else show, you can do it with less cuts.

2020/01/27 10:48 PM
He can cut off 50 there will be 55 left
then cut off another 50 there will be 10 left
then cut off 10
the minimum number of cuts is 3