Is there a time lag in the action of interlocked cogs? Consider the following scenario.
A central cog turns a connected cog and supplies at least some energy to balance frictional losses. The cogs mesh perfectly and there is no slippage. If another cog is added on the end, a little more driving force is needed, but the new cog still meshes and moves in sync. In this way, a chain can be constructed of indefinite extent. Consider such a very, very long chain with all the cogs moving together. The first and last cog are synchronous. Does this mean the influence of the driving cog on the end cog exceeds the speed of light? Explain. What if you assume the material the cog is made from is incompressible?
Cogs often feature in recreational mathematics, a field that has been evolving over millennia and is still developing. There are key ideas and principles in recreational mathematics that recur and are reworked into new forms. And of course, there are innovations, brand new ideas that emerge such as ‘Cheryl’s birthday’. However, such innovations are pretty rare because of the difficulty in thinking of new ideas. It is much easier to tweak existing puzzles than to make up new ones. A classic puzzle is the ‘think of a number’ trick. Here is a variant:
Think of any number, whole number or fraction, positive or negative, and do not reveal it. Multiply the number by 3. Then subtract any number you like, but you need to say what that number is (assume here it is 22). After subtraction, multiply the result by 3. Repeatedly add all the digits of the final number until it reduces to a single digit. That number, you are correctly informed, will be 6. Explain how the trick is done.
Very common also are puzzles of this type: ‘Mr and Mrs Smith have two children; at least one is a girl; what is the probability one is a boy?’ These problems tend to be confusing and have unexpected answers because there is a hidden cheat. You are actually given ‘second-hand’ processed data where there is already knowledge of the results before the ‘reveal’. In this example, the normal randomness associated with the statement, ‘Mr Smith and Mrs Smith have two children’, is lost. The Smiths are not a random family with two children – the Smiths better represent the subset of families with two children where at least one child is a girl.
The Smith family have four children. They decided from the start their family should be made up of an equal number of boys and girls, and this they have achieved. How did they manage it?
Packing puzzles are also popular. Try this variant.
Two criminals break into a bank vault and fill a cube-shaped crate they brought with them with gold ingots. Once the crate is completely full with no spaces at all, the lid is nailed down. They lift the sealed crate into the back of their van and drive off. The crate will go into storage for a year till the heat dies down. If the ingots are exactly 5 cm x 3 cm x 1 cm in size, calculate how much gold they stole? Show
the answer is unique.
You can find the average distance or the average time by the process of adding a set of distances or times and dividing by the number of data points, but the same does not apply to velocity because it is a derived quantity. There are a number of puzzles based on the natural assumption that velocity values are simply averaged. For example, ‘Jack cycles 5 miles at a speed of 15 miles per hour and cycles back at a constant speed of 10 miles per hour. What was his average speed?’ The answer is not 12.5 miles per hour, but is instead 12 miles per hour.
Samantha is a very competitive cyclist and regularly cycles to a beach 15 miles away. The ground is flat and she can maintain a constant speed of 15 miles per hour there and back on a calm day. This day there is wind of steady 5 miles per hour blowing from the same direction for the duration of the two-way journey. Should Samantha be disappointed the cycle has taken her an extra 15 minutes?
Election problems can be tricky. In the example below, it seems like there is not enough information, but it is much easier than you might think.
In the 2016 and 2021 Scottish parliamentary elections, a particular seat was contested only by the Scottish National Party (SNP), the Scottish Labour Party (SLP) and the Scottish Conservative Party (SCP). The seat changed hands in 2021 with the SLP vote increasing by 5%, the SNP vote increasing by 10% and the SCP vote decreasing by 20% (all with respect to the 2016 counts). Who won each election and did any party lose their deposit (triggered if a candidate gains less than 5% of the vote)?
Probability questions make popular puzzles because of the tendency to attack them using intuition when that is the very last thing that should be done when faced with a probability calculation. The next task presented below is not really a puzzle but is a problem that tests your understanding of probability. The analysis required is very difficult and as far as I see cannot be solved without using calculus.
Philip is leaving for college to study mathematics. His father tells him that he will be given an allowance of £100 a week for food and entertainment. However, to encourage saving he is told that each week he will be given a random amount between £0 and £200 instead of a fixed sum. Philip protests this is unfair because there could be several consecutive weeks with very little money. He proposes setting a threshold value to avoid this possibility. If the amount on a particular week is W and is above the threshold value L, he will be given a random value between 0 to W the next week. If it is below, he will be given a random value between W and 200 instead. This will act as an inhibitor on long sequences of high or low values. Philip goes on to say, ‘As our house number is 121, I suggest that should be the threshold figure’. His father agrees to this. Does this plan alter the average amount he will receive and, if so, by how much?
Some puzzles can involve some very basic physics, but the principles need to be elementary for the problem to be solvable for the majority. Look at the example below which seems like it needs a lot of specialised knowledge, but doesn’t.
A prestigious golf complex has a number of golf courses spread over an area of 10 km2. Some club members refuse to play when there is a chance of lightning. A physics student was engaged to identify the danger spots. She installed a 10 m pole at each end of the facility. On top of each was placed a UV light sensor triggered by light from any direction (but not registering the direction), a microphone, and a battery-powered data logger. The two loggers have independent clocks. After a year, the data was retrieved from each and an attempt was made to place each strike by triangulation. What mistake did she make and how did she correct it the following year?
Or a basic knowledge of chemistry might be needed.
A class watch from a distance as a technician mixes chemicals to make a gas. The experiment is repeated each week for 6 weeks. The students notice that the reaction ends about 5 minutes earlier on occasions the technician is not wearing a lab coat. Why?
And finally, some nonsense.
Why, mathematically, could women be considered one-dimensional whilst men are two-dimensional?