The Sand Reckoner of Archimedes

Archimedes of Syracuse was a Greek mathematician who lived in the 3rd century before the common era. He was probably the greatest mathematician of the antiquity

Among many other things, he was interested in devising a notation for very large numbers. In order to present his suggestion for a system to represent very large numbers, he proposed the following problem:

How many grains of sand would fit inside a sphere as big as the Universe?

Archimedes of Syracuse, Psammites (The Sand Reckoner), 3rd century BCE

For the size of the universe, he used the heliocentric model of Aristarchus of Samos, and estimated (well, took a wild guess) that the sphere of fixed stars has a diameter of (what we would call today) about 2 light years. He then assumed that a sphere with a diameter of one Greek inch (about 19 mm, or 3/4 of an English inch) can fit 640,000,000 grains of sand. (He was interested in a system for representing very large numbers, after all.)

He finally reached the conclusion that the Universe could fit no more that what we would write today as 10⁶³ grains of sand.

The interesting thing is that he actually devised a system for writing such a large number. In the 3rd century before the common era.

Where a cannon ball will land. A problem of extreme importance in the Middle Ages, why Newton and [Galileo][1] were studying gravity, and (because of the scale involved) nearly impossible to estimate with the precision desired by commanders.

Because of the great mass of a cannon ball, the effect of wind and complicating factors of air resistance are significantly reduced.

The equations of motion are a calculation that provides an exact error. Because, as others have pointed out, the calculation is incomplete, there is a built-in error.

Since you've drawn a distinction between 'estimating' and 'calculating', I think estimation would involve spotter balloons and a network of relay flagmen to communicate where a first shot landed, and the spotters giving guidance to 'walk' the fire towards the desired target.

You can maybe see why estimation wasn't beloved by commanders. It required being the first to the field with enough lead time to set up such a network, the chain didn't cover a great deal of field, was easily disrupted by weather and enemy action, and was tremendously difficult to quickly move if the action was happening somewhere else.

[1][http://galileoandeinstein.physics.virginia.edu/lectures/gal_accn96.htm]

Estimate exponential growth using grains of rice and a chessboard.

The story goes:

The ruler or India was so pleased with one of his palace wise men, who
had invented the game of chess, that he offered this wise man a reward
of his own choosing and he said to the man: “Name your reward!”

The man responded: “Oh emperor, my wishes are simple. I only wish for
this:

-Give me one grain of rice for the first square of the chessboard, two grains for the next square, four for the next, eight for the next and
so on for all 64 squares, with each square having double the number of
grains as the square before.“

The emperor agreed, amazed that the man had asked for such a small
reward – or so he thought. After a week, his treasurer came back and
informed him that the reward would add up to an astronomical sum, far
greater than all the rice that could conceivably be produced in many
many centuries!

The total number of grains would in the Emperor in the story's estimation be quite manageable - but with precise calculation turns out to be: 18,446,744,073,709,551,615 - purported to be sufficient rice to cover the whole of India's landmass a meter deep in rice.

The math is straightforward, it involves the addition of a series of numbers from 1, then 2, then 2², then 2³, 2⁴... and so on up to 2⁶³. (Ie. the first square = 2⁰ aka 1).

Date of the next lunar/solar eclipse. See “Antikythera mechanism”:
https://en.m.wikipedia.org/wiki/Antikythera_mechanism

Eclipses follow such a complex pattern, they cannot be estimated from previous events, but as the Greek mechanism shows, they could be calculated.

Decrypt an RSA-style Message

Although the specifics of RSA encryption and decryption - or even more generally public key cryptography didn't come around until the 1900s, the ideas of factorization go back to the Greeks.

Furthermore, RSA itself works on rather simple mathematics: multiplication and modulus operations. Wikipedia even has a simple example that you can do by hand yourself. These were definitely available in the middle ages. There's no reason why this wasn't done in the middle ages except that (a) it's tedious when done by hand, and (b) no one came up with it. If Mr. P is thoughtful and ahead of his time, there's no reason why he could not have come up with a scheme like this. As an aside, there were many attempts in the middle ages to hide messages (ways of folding paper, using secret codes, created locks and boxes, etc), so there was definitely a desire for this type of thing.

Anyway, Mr. P could explain clearly how the algorithm works, give the private key and encrypted message, and then ask Mr. E to uncover the original source. If Mr. E is not exactly accurate, then the result will be garbled text / a wrong number. This demands precision.

Tides

Anything that can be calculated can be estimated. However something with a sinusoidal function like the tides requires significant precision even on an estimate, if you're betting on ebb and you get flow, or you're betting on a high tide and you get low, you're in a reasonable amount of trouble. There's no slack for adding a margin for error, you're right or wrong.

This has an added complication in that time is also a fairly approximate matter in the period. So asking what time high tide is wouldn't work, but asking for the state of the tide at sunrise on the morning of a specific date should be good enough for anyone as it requires both the patterns for the movement of the sunrise as well as the tide.

Anything with which neither party has experience

The key to a good estimate is that you have to know a fair amount about the situation. You might know how much food an army needs for a week or how many cannonballs are required for a campaign. You might know how long it takes to paint a wall of any size and how much paint is required. You might know how many litres of water are in an arbitrary swimming pool or the fuel consumption of an average small car. But if you know nothing about the subject then it's not possible to make an estimate.

The Archimedes eureka problem.

Have a number of strangely shaped objects made out of different materials.

The challenge is to work out which of them is made of the densest material.

You can't estimate that as they are strangely shaped. You can't just weigh them as they are different volumes.

The solution is to weigh them, then sink them in water and see how much the water rises. This lets you work out the volume. You can then divide the weight by the volume for each to get the density.

This was how Archimedes worked out whether a crown was made of gold or not.

A simple, trivial, one would even say childish. Calculation of how much of a square foots the kingdom is. While knowing it's two dimensions.

Your Mr. P is John Napier. Who wrote a book called

Description of the Wonderful Rule of Logarithms

Using word Wonderful was just to smear in the face of Mr. E (of whom history forgot) that John found a way to calculate faster and with very high precision.

I, the King, wish to share the Kingdom's wealth with the People. If the Kingdom's population keeps growing, how long before they collectively are richer than the Royal Family?

An estimate would say 'Probably 100 years'. An exact formula says never.

Stick with me here.

Let's say this is a verrrryy nice king. What goes around comes around- he shares his wealth with his people. For every sum of cash(for ease, we're going to call this amount $2C$) that comes into the kingdom, he takes $frac12C$ for himself. He then decides to group the kingdom by population and closeness to himself: every group has one more person than the one above it. He's on the top by himself, his Queen and heir are one below, his three knights below that, four nobles, etc., all the way down to his thousand peasant farmers. Everyone in each tier gets the cash divided up like this:

So, the fraction of C you get is one over two to the power of the number of people in your tier.

The rich stay rich and the poor stay poor, but everyone's grateful to the king because he gives each of them enough to sustain their own lifestyles, but critically not enough for anyone to move up or down. When you're born, you're shoved into the 'lowest bottom pile'. If someone dies higher up and you have a right to that space, it's yours and you get the cash. For this reason, this system can scale towards infinity.

His Royal Highness is also very intelligent. He knows money talks, and is aware of the meltdown that could occur if he suddenly became very unpopular and the people had more wealth than he did. Or he's an egomaniac. Either way, he wants his family to be #therichest.

With this system, what's the maximum population size you can have before the people have more wealth than the king?

An estimate would say "Well, given the current population growth, I'd guess 20 years your Highness?". However, a Medieval mathematician (Oresme) proved that this sequence converges to 2. Hence why I used $2C$ at the start. Read that proof. It's truly brilliant.

Finally, some simple math:

$Wealth_King = frac12$

$Wealth_Queen + Wealth_Heir = frac14 + frac14 = frac12$

$Wealth_Royal,Family = 1$

$Wealth_Infinite,population = 2-1 = 1$

$Wealth_Infinite,population = Wealth_Royal,Family$

So, as long as the population is less than infinite, the King can share the Kingdom's wealth with his people and his family will always be richer than everyone else put together. Neat, right?

I'm aware this is such a botched explaination of pretty much everything, so let me know if something's unclear and I'll try to clean it up in an edit.

Hope it helps!

The Problem of Proving

Any test they would have to do needs a proof. Its nice that MR. P would calculate the exact right value and Mr. E guesses exactly the same value, both might be wrong and nobody would know it. So you need something that is not know, hard to estimate but possible to measure or reassure. Problem is that some calculations might require information that nobody has or can aquire and so the expected result is different. So everything with random elements like natural occurences are out. All we can do are inarguable facts, that can be meassured.

The Size of the Earth

Back in around 200 BC Eratosthenes calculated the circumference of the earth. He knew that a certain time in the year the sun would not cast a shadow in a well at noon in Syene. He knew the distance to Alexandria, that was directly north and asked a friend to measure the angle a shadow would cast from a stick. With that angle and the know distance, he was able to calculate the earths circumference.

Problem is, Mr. P or Mr. E could know that (although unlikely). On top of that, in the middle ages, noone could really measure the distance, so whatever.

But this value could also be used to calculate different values given the angle of shadows at their current position. For example what the angle another sticks shadow would show that is 100 miles further north or south from their current position. Or the distance to a to a random location north or south of their current position given the angle of the shadow there and their current location. If it is a place both dont know, but the distance can be measured, then Mr. P should be able to calculate the distance, but Mr. E should not be able to guess it.

The Birthday Paradox (simplified).

Have six jars, each with twenty numbered balls (the ten jars can hold differently coloured balls to make sorting simpler), 1 to 20.

Say we extract one ball from each jar, thus obtaining a collection of six balls.

How often will (at least) two of those balls share a number?

Solution:

Calculate the probability of this not happening. The first ball can be any number. The second ball must get one of the remaining 19 numbers, which will happen 19/20ths of the time. The third ball will come up right 18/20th of the time. The fourth will do so 17/20th of the times. The fifth and sixth, 16/20 and 15/20. So all of them will come up right with a chance of (19*18*17*16*15)/(20*20*20*20*20), which is 0.43605, or 43.605% of the time.

Therefore, 56% (more than half) of the time two balls among six will sport the same number.

In the original formulation, "how often in a class of 23 children at least two will share their birthday" (p1=1/365.2544) will also yield around 50%.

To prove who's right, they can design a chance game - extract six balls, bet that there will be two with the same number; if you're right, you get X times the bet. Both the mathematician and his opponent can "offer" a value for X, whoever offers the smallest value of X and manages to be ahead after 20 tries is the winner (the more the tries, the less luck enters into the matter).

For example, the mathematician offers 1.80 dollars to the dollar (which is 1/0.43). 43% of the times he will lose, 56% of the time he will win 180% of the bet; in the end he will have gained S*56%*180% = S*0.56*1.80 = a bit more than S. His opponent might offer double the stake, which will make him lose since 1.80 is less than 2.0; or he might offer one dollar and a half per dollar, which will make him lose on average 56%*150% = 16% of the time. After twenty tries, the chances of him being ahead are negligible (0.84 to the 20th power is about three per cent).

What if instead of having to calculate something very precisely, you proposed a MQC (Multiple Choice Questions) instead?
A MCQ is a list of questions with (most of the time) 4 proposed answers given for each.

The sky is :

• A. Blue


• B. Solid


• C. Brown


• D. Above our head

To make it even trickier, sometimes MCQs have malus points if you answer something wrong. That way you can't just randomly pick something when you don't know.

You can't estimate the answers of a MCQ. It's either A, B, C or D.
If Mr P is really that much greater than Mr E, he'll get only right answers, whereas Mr E will lose points by lacking accuracy. You could even ask each one of them to make a test for the other one; as long as he's better, Mr P will always win :)