MathDB

A Physicist for Fun discovered three types of very peculiar particles, and classified them as

P

H

and

I

particles. After months of study, this physicist discovered that he can join such particles and obtain new particles, according to the following operations:
• A

P

particle with an

H

particle turns into one

I

particle;
• A

P

particle with an

I

particle turns into two

P

particles and one

H

particle;
• An

H

particle with an

I

particle turns into four

P

particles;
Nothing happens when we try to join particles of the same type. It is also known that the physicist has

22

P

particles,

21

H

particles and

20

I

particles.
(a) After a finite number of operations, what is the largest possible number of particles that can be obtained? And what is the smallest possible number of particles?
(b) Is it possible, after a finite number of operations, to obtain

22

P

particles,

20

H

particles, and

21

I

particles?
(c) Is it possible, after a finite number of operations, to obtain

34

H

particles and

21

I

particles?

1

Problems(2)