## Sunday, January 17, 2016

### Bitter pills (some unfortunate math)

During a hike in the south, J1 tripped and cut his forehead badly. This incident lead to the following conversation snippets and math observations about taking medicine.

# How many ways

J1 has to take three different pills, a white one, an orange one, and a cream one. How many ways can he take them? 6, of course, 3 choices for the first, 2 for the second, and then whatever is left at the end. Follow-up: why do we multiple the choices instead of adding them?

However, our answer is 12. The twist is that one pill is broken into two halves because it is so big. That means we have 4 pieces, so 4! ways of arranging, but two pieces are identical, so 4!/2 distinct ways of ordering the pieces.

For those who are more advanced: what if we allow taking more than one pill at a time?

# Cut in half and comparisons

As noted above, the white pill is pretty large, 375 mg. We break it in half, so how large is each half?
The next largest pill, the orange one is 25mg. How many times larger is the white compared with the orange?

The smallest, the cream one, is 5mg. How many times larger is the orange? How many times larger is the white?

# Some pictures

Sorry to disappoint you, if you are looking for pictures of the wound. It really is too gruesome to spring on the math lovers of the internet.

Not our favorite ten frames:

Which sequences of pattern blocks close up?

Peg Board Fun

# Primes and Powers

Saw this tweet and did a short investigation with J2.

His immediate reaction was a great one: "Let's try 2 and 3."
For uniqueness, we verbally talked through 5 and 3, but I guided him to recognize that these would both be odd. That led him to recognize that the only possibilities were an odd prime and 2, so we worked through some more cases and came up with a related conjecture: