A link between Ramanujan and Fermat near misses

My latest book ‘The Simpsons and Their Mathematical Secrets’ explains that two episodes of ‘The Simpsons’ contain references to Fermat’s Last Theorem. In fact, the episodes contain so-called near misses, which are sets of numbers that almost fit Fermat’s notorious equation, but not quite.

The book also discusses ‘Futurama’, and explains how 1,729 appears in several episodes, because the mathematician Ramanujan commented that it is the smallest natural number that is the sum of two cubes in two different ways.

1,729 = 103 + 93

1,729 = 123 + 13

Of course, there is link between Fermat near misses and Ramanujan’s 1,729, and therefore a link between the mathematics of ‘The Simpsons’ and the mathematics of ‘Futurama’, because:

Fermat’s Last Theorem looks for solutions to          xn + yn = zn
where n>2.

The two ways to form 1,729 can be matched as   103 + 9= 123 + 13

In other words (103 + 9= 123) is a near miss solution to Fermat’s Last Theorem, as it only misses by 1 (or 13).

I am grateful to Mike Hirschhorn, a mathematician at the University of New South Wales, who pointed out that Ramanujan identified a way to generate an infinite number of near misses of the form:

x3 + y= z3 ± 1

 If you want to find out more, then you can read Mike’s papers on Ramanujan and Fermat near misses on his website (39, 40, 107 and 128).

 

 

David X Cohen – Puzzle 2

Competition closedanswer here.

Thanks to everyone who entered the previous David X Cohen puzzle competition – there was such a great response that here is another one.

Once again, this cryptarithmetic puzzle substitutes each letter for a number in order to make the multiplication below valid. This puzzle is based on the name of the great David X. Cohen (a writer on The Simpsons and co-creator of Futurama), such that his middle initial represents a multiplication sign. The nine different letters in DAVID COHEN represent nine of the digits 0 to 9 (which means it is trickier than the previous puzzle, as it was restricted to the digits 1 to 9).

DAVID x COHEN = 250266547

Competition closedanswer here.

Big thanks to James Grime (@jamesgrime), who invented this puzzle.

David X Cohen Puzzle 1 – solution

Congratulations to Steve Everson, who won a copy of “The Simpsons and Their Mathematical Secrets” by correctly deciphering:

DAVID x COHEN= 763860049
as
13241 X 57689 = 763860049

He summarised his method as: “Started from the lowest digit (there is only one assignment for D and N that works) and worked my way up.”

In other words, obtaining 9 as the final digit of the product means that the final two digits of DAVID & COHEN have to be 3 & 3, or 7 & 7, or 1 & 9, or 9 & 1. However, 3 & 3 and 7 & 7 cannot be correct, as the digits have to be different. Similarly, 9 & 1 cannot be correct, because if D=9 then it both ends and starts with a 9, and you that would force COHEN to be a 4-digit number, when it has to be a 5-digit number – does that make sense?

Once you have got as far as 1AVI1 x COHE9 = 763860049, then you can pick away at the rest of the number.

Alternatively, I know that some people used a brute force computer search approach. Moreover, some guessed that the solution might be a pair of primes, which made searching much easier.

 

 

David X Cohen Puzzle 1

Competition closedanswer here.

This cryptarithmetic puzzle substitutes each letter for a number in order to make the multiplication below valid. This puzzle is based on the name of the great David X. Cohen (a writer on The Simpsons and co-creator of Futurama), such that his middle initial represents a multiplication sign. The nine different letters in DAVID COHEN represent the digits 1 to 9.

DAVID x COHEN = 763860049

Competition closedanswer here.

Big thanks to James Grime (@jamesgrime), who invented this puzzle.

If you are new to cryptarithmetic puzzles, then here is a primer.

How Ptolemy’s theory of epicycles can explain anything – including Homer.

Ptolemy’s theory of epicycles (orbits within orbits within orbits…) was used to explain the strange motion of the planets, which sometimes flipped back on their own paths, instead of following simple patterns. At the same time, crucially, his theory allowed the Earth to remain the centre of the universe.

The problem is that Ptolemy’s theory can be used to justify any set of orbits, because the epicycles can be adjusted to describe any path. Indeed, a sufficiently complex and honed set of epicycles can even describe a planetary path that draws an outline of Homer Simpson.

 

More about epicycles here and in loads of other places.

Al Jean – mathlete

A photograph of the mathematics team from the 1977 Harrison High School yearbook. The caption identifies Al Jean as the third student in the back row and notes that he won gold and third place in the Michigan state competition. Jean’s most influential teacher was the late Professor Arnold Ross, who ran the University of Chicago Summer program.
1.3 Al Jean

Mike Reiss – the mathlete

Mike Reiss (second in the back row) on the 1975 Bristol Eastern High School
Mathematics Team. As well as Mr. Kozikowski, who coached the team and
appears in the photograph, Reiss had many other mathematical mentors.
For example, Reiss’s geometry teacher was Mr. Bergstromm. In an episode
titled “Lisa’s Substitute” (1991), Reiss showed his gratitude by naming Lisa’s
inspirational substitute teacher Mr. Bergstromm.
1.2b BEHS_Math_Tam_1975-small