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Blue mountains at Katoomba

Welcome

The Katoomba Men’s Shed is a not for profit community organisation affiliated with Australian Men’s Shed Association. It aims to promote the health and well being of men, and provides an opportunity for men to share time together,  swap yarns and work together on community projects.

The Men's Shed movement began in Australia in the 1990s. There were concerns about men's health, but there was little encouragement in Australian culture for men to meet and talk over their issues. However, men were comfortable with the concept of the backyard shed, where many would go on a regular basis to potter about and work on projects. So the concept was born of a community-based shed as a place where men could go to meet other men, socilise and work on projects.

Solutions to puzzles from August 2017 edition of Noise from the Shed

The census

The census collector comes into a house and asks: ‘how many people live in this house, and what are their ages’. The householder says: ‘I live here with my three children. I am 36 years old. If you multiply the ages of my children, the result is also 36. If you add the ages of the children, the result is the same as the number of windows in the building across the road’.

The collector looks at the building across the road, and after a moment of thought says: ‘I can't work out your children's ages from those clues, I need more information’.

The householder says: ‘The oldest child has red hair’.

The collector says: ‘Thank you, I now know the ages of your children’.

What are the ages of the three children and how many windows are there in the building across the road?

The solution

List all the possible combinations of three numbers that when multiplied result in 36, then add them up:

  • 1+1+36=38
  • 1+2+18=21
  • 1+3+12=16
  • 1+4+9=14
  • 1+6+6=13
  • 2+2+9=13
  • 2+3+6=11
  • 3+3+4=10

The Census collector knew how many windows there were in the house across the road, but could not use that to select the right answer, so we can deduce that there must not have been a clear answer. There are two sets of ages that add up to 13, so that means there must be 13 windows. If it was any one of the other numbers, the census collector could have worked it out.

And then the householder said that the oldest child had red hair. This statement means that there is an oldest child. So the only possible answer is 2, 2 and 9.

Therefore the ages of the children is 2, 2 and 9.

Pills

Your pharmacist gives you 5 jars filled with an identical number of pills. The pills contained in each jar have an identical appearance and size, so if you mixed all of the pills you wouldn't be able to tell them apart.

4 of the 5 jars hold pills that weigh 10 grams each, and the remaining jar holds pills that weigh 9 grams each. However, you don't know which jar is the one holding the lightweight pills.

You have a set of electronic scales that are accurate to within half a gram, but you only have enough battery power for a single use.

Using the scales only once, how do you determine which jar is holding the 9 gram pills? (Assume there are more than enough pills for your needs.)

The Soluton

Take a different number of pills from each jar and compare the expected mass with the observed mass. For example, take:

  • 1 pill from jar 1
  • 2 pills from jar 2
  • 3 pills from jar 3
  • 4 pills from jar 4
  • 5 pills from jar 5

Expected mass (if all pills were 10 grams each) = (1+2+3+4+5) * 10 = 150 grams.

Suppose jar 4 contained the 9 gram pills. The total mass that you observe would be (1+2+3+5) * 10 + 4*9 = 146 grams. Since the mass you observe is 4 grams less than the expected 150 grams, you know jar 4 is holding the 9 gram pills.

In general (using the distribution of pills from above):

Jar number = Expected mass (150g) - observed mass

Another, more elegant, method is to take 9,8,7,6,5 pills from each jar respectively. The last digit of the weight will be the number of the jar with the lightweight pills. (The sample is 35 pills. If the total weight is 344 grams, then jar 4 has the lightweight pills.)