The Glass Bead Game

Herman Hesse’s 1943 book, “The Glass Bead Game” tells the story of Joseph Knecht, the ‘Magister Ludi’ (Master of the Game). This Glass Bead Game is not fully described in the book but essentially is :-

“an abstract synthesis of all arts and sciences. It proceeds by players making deep connections between seemingly unrelated topics”[Wikipedia]

Inspired by this idea, we set out to make a real Glass Bead Game that captures the essence of Hesse’s concept. Below are the results of our efforts.

This quasi-Glass Bead Game(GBG) is best described by an example :-

Example 1: Traverse from Miles Davis to Big Bang via Blade Runner.

Meaning, we have to somehow connect these concepts together in a meaningful way, perhaps using intermediate steps. For example, an initial step might be :-

Step 1: Miles Davis → Kind of Blue

A reasonable connection since Kind of Blue was a major album of Miles Davis. The second step could be :-

Step 2: Kind of Blue → Deep Blue

Deep Blue was the computer chess program that beat the World Champion Garry Kasparov in 1997. The ‘Blue-Blue’ connection seems a little tenuous but let’s continue.

Step 3: Deep Blue → Artificial intelligence

This seems ‘acceptable’ given that Deep Blue is certainly an important example of A.I.

Step 4: Artificial intelligence → Blade Runner

If you're familiar with the ‘Blade Runner’ film/book, it is based on the concept of artificial beings called replicants.

Step 5: Blade Runner → Off-world Colony

Again, a reference to the book - people can go ‘off-world’ to escape the dying Earth.

Step 6: Off-world Colony → Cosmos

Off-world is in space. Again a little tenuous but let’s finish the GBG anyway..

Step 7: Cosmos → Big Bang

Because the Cosmos was started with the Big Bang.

So we now have a completed GBG :-

Miles Davis → Kind of Blue → Deep Blue → A.I. → Blade Runner → Off-world Colony → Cosmos → Big Bang

The skill in the game comes from making interesting, intelligent connections between the topics and also through using as few terms as possible, whilst trying to retain the elegance of the solution.

You may have your own opinion on whether the above GBG solution is interesting, intelligent or elegant but I think it is fair to say that it is a valid solution to the original GBG. At the same time it is clearly in no way optimal. How would you improve it?

Is it possible to score a GBG?

After playing around with a number of GBGs and discussing them at length we found that the games were certainly fun to attempt but we were left wondering about how we could ‘score’ a solution. i.e. Is the quality of a solution inevitably subjective?

Take the GBG below and two solutions :-

GBG: Traverse from L'Ortolan to Great Wall of China via Time Machine

Solution 1: L’Ortolan → Michelin guide →Travel Guide → Time Machine → Morlocks → Communism → China → Great Wall of China

Solution 2: L’Ortolan → Michelin → Pirelli → Calendar → Time Machine → 1960 → Great Chinese Famine → Great Wall of China

Which is the better solution? And how can one make this judgement? Some of the links may feel too obscure in your opinion or even not understandable. Can a link be ‘illegal’?

These were the questions that we pondered. But after some time, we stumbled upon an idea that gave us the beginnings of an objective way of scoring GBGs

The Internet comes to the rescue

The essence of the solution comes by using the power of the Internet - specifically Google searches.

Here’s how the method works :-

Step 1: Run a google search on each term in the solution

e.g. “L’Ortolan” then “Michelin Guide” etc and make a note on the number of hits for each search. Note that the quotes are very important since Google should treat “Michelin Guide” as one item rather than two.

Step 2: Run a google search for each link

e.g. “L’Ortolan” “Michelin Guide” together in one search. Then google “Michelin Guide” “Travel Guide” etc. Make a note of the number of hits for each search.

Step 3: Calculate the score of each link and total them

The score of each link is LinkSearchHits / (Item1SearchHits + Item2SearchHits)

e.g. Hits for ‘“L’Ortolan” “Michelin Guide”’ divided by (Hits for “L’Ortolan” + Hits for “Michelin Guide”)

Add all such link scores together

Step 4: Calculate the final score

The Final Score is 10000 * (LinkScoreTotal / (NumberOfItemsInGBGSolution Squared)

Sample Calculation

Let’s do a sample calculation for both our solutions.

First of all, Solution 1 :-

Solution 1: L’Ortolan → Michelin guide →Travel Guide → Time Machine → Morlocks → Communism → China → Great Wall of China

Step 1: Run a google search on each term in the solution

Search Item Number of Hits

“L’Ortolan” 3,750,000

“Michelin Guide” 9,710,000

“Travel Guide” 107,000,000

“Time Machine” 30,900,000

"Morlocks” 4,390,000

“Communism” 2,490,000,000

“China” 4,410,000,000

“Great Wall of China” 5,070,000

Step 2: Run a google search for each link

Linked Items Number of Hits

“L’Ortolan” “Michelin Guide” 1,370

“Michelin Guide” “Travel Guide” 209,000

“Travel Guide” “Time Machine” 1,270,000

“Time Machine” “Morlocks” 93,700

“Morlocks” “Communism” 47,900

“Communism” “China” 16,100,000

“China” “Great Wall of China” 6,410,000

Step 3: Calculate the score of each link and total them

Remember that the score of each link is LinkSearchHits/(Item1SearchHits+ Item2SearchHits)

Linked Items Link Score

“L’Ortolan” “Michelin Guide” 1,370 / (3,750,000+9,710,000) = 0.0001

“Michelin Guide” “Travel Guide” 209,000 / (9,710,000+107,000,000) = 0.0018

“Travel Guide” “Time Machine” 1,270,000 / (107,000,000 + 30,900,000) = 0.0092

“Time Machine” “Morlocks” 93,700 / (30,900,000 + 4,390,000) = 0.0027

“Morlocks” “Communism” 47,900 / (4,390,000 + 2,490,000,000) = 0.0000

“Communism” “China” 16,100,000 /( 2,490,000,000 + 4,410,000,000) = 0.0023

“China” “Great Wall of China” 6,410,000 / (4,410,000,000 + 5,070,000) = 0.0015

Total = 0.0176

Step 4: Calculate the final score

The Final Score is 10000 * (LinkScoreTotal / (NumberOfItemsInGBGSolution Squared) =

10000 * 0.0176/(8*8) = 2.75 (to two decimal places)

So what is the score for Solution 2?

Solution 2: L’Ortolan → Michelin → Pirelli → Calendar → Time Machine → 1960 → Great Chinese Famine → Great Wall of China

Instead of doing it completely manually, we can use this spreadsheet to automate the calculations.

By doing so, we get a score of 1.79

This implies that solution 1 is superior. Do you agree?

What are the advantages of this method?

- It rewards strong connections between links in the GBG

- It discourages the use of very common terms

- It discourages long chains of links

For these reasons, we believe that the method is a reasonable way to objectively score a GBG solution. Perhaps it can be improved? - tell us how!

Meanwhile, have fun challenging friends and family with GBGs!