Ron Barnette's Zeno's Challenge #43 result

Dear Zeno's patrons: Here are the results....very creative suggestions...over 350!

The Coffeehouse owners, Charles and Maggie, have been concerned that their excellent soda bread, a favorite with the patrons, might be depleted early on St. Patrick's Day, leaving many other bread lovers disappointed. A crafty old patron, Gilbert, smiled and posed the following theoretical solution to their quandry:

"My dear Charles and Maggie, please hear me out as I formulate a potential solution, in the very spirit of Zeno himself. Bake your excellent loaf of soda bread, and serve it to those who want SOME, with the following instruction for all who take bread: Take a portion, less than half of what is given, and then allow others to do the same with the remaining portion. Then divide your portion in half, and present the remainder to still others with the same instruction. Admittedly, some bread-lovers will get larger portions than others, but, THEORETICALLY, there's enough bread portions (no matter how small) for EVERYONE if these instructions are followed! Think about it."

Wow! Gilbert has made quite a claim indeed. IN PRINCIPLE (ignore the practical problems entailed), is Gilbert correct??? Talk about loaves and fishes!!!!! Reflect on this, and submit your answer to our latest Zeno's Challenge....Ron Barnette, Zeno's proprietor

It seems that the responses were quite divided on this thought-provoking challenge, and I will leave it to your discretion as to which side reasoned the best answer. As such, I have included examples from both the 'yes' and the 'no' camps. And to all respondents, I wish to share my thanks for your logical time invested! Zeno's continues to be a fun and challenging meeting place...RB

Tony Stewart..country: Sweden
comments: Well, yes, Gilbert is essentially correct in the same way Zeno himself was in his famous dichotamy in which a runner never reaches the final goal because he must first reach each halfway point before reaching that goal. The distance of the run can always be divided into half again, thus leaving the runner with another partial goal to complete...and another and another... Granted the "halfways" become infeasibly small, but the runner must always get halfway first. He never gets to the final goal.

Now, with the Tasty St Patrick's cake, the idea is much the same(is the cake green by the way)? One can divide something in half forever. 1/2 a loaf first(best to be at the bakery early i suppose) then 1/4 loaf, then 1/8 loaf, then 1/16, 1/32, 1/64 and so on and so forth...until, well, until nano surgical tools are needed to cut up the halves, and then it doesn't stop there. The halves simply become infinitely small. But, everyone gets some. So, Charles and Maggie do not need to fret about faulty advertising and lawsuits.
No need to worry for there is always enough to go around.
If they follow Gilbert's advice that is.

Karen Etling...city: Los Angeles
state: CA
country: USA
comments: NOTE: This is an editorially proofed revision of the same response sent less than an hour ago. <blush> Sorry 'bout that:
Short reply:
Yes, there will be enough bread for all, however *infintessimal*. ;-)
Detailed reply, replete with a quasi-logical proof:
The Explanation
This is a thought problem derived from a mathematical limit (of two finites, where one, and only one, approaches Infinity). To apply it to the problem presented, think of the broken bread as two finite objects, labelled for the heck of it as X and Y. Ok, so let's call the bigger piece "X" (X can be of any size, but greater than 1/2 the entire, pre-broken, loaf.

The other piece, "Y", is the one to be broken (or halved) so that all will receive a piece of this part of the loaf. So let's have at it: Halve the "Y" loaf-piece, half it again, and again, and so on ... ad infinitum. (Or, as Y--> infinity.) Hence, the pieces of "bread", halved an infinite number of times, will never reach 0 (or, in this case, the bread will never run out...*ever*). Though it's not pertinent in this example, that other piece, "X"? Meanwhile, it's the one approaching some big piece 'o loaf, and why we don't just use that, <shrug>. (Theoretically we can, but we have to introduce, via calculus, anyway, another dimension that only confuses this already confusing explanation. Suffice it to say, it's possible, *but not in "halves"*.)
An aside:
Ok, so at this point I tried to write out in long-hand of the mathematical concept of a limit (a "limit" is, by the way, a heart of Newton's theory of gravity, and of spacetime, and other practical matters of this earth, (e.g. a given formula for a bridge that won't collapse under force, along with the engineer who designed it). This is, through various forms of math (I used calculus in my example), how we (even mathematicians and scientists) do *most* of their applied work. It's the quantum and astro physicists et al, some theoretical mathematicians, and a few philosophical nuts <g> who use Einstein's Relativity theories (both relating to gravity, though the second only using the first as a "given, thus far more profound) to study such things as spacetime, singularities (blackholes), curvature of time (and with it, space) of the cosmos, and discovering that what we perceive on this earth ain't necessarily the case in the Universe. I only mention this for any budding philoso!
phers who haven't yet taken Phil of Science (or did, but your prof didn't want to "go there"). Either/or, do some pre-class prep and blow your prof away. Heheh... :) C E: DO NOT do this if you expect a letter of ref for grad school!!!!!!!

Jen Walker (age 15)...city: Sarnia
state: ON
country: Canada
comments: Gilbert's claim is most definitely correct. There is no possible way to finish the loaf. You would first have 1/2 of the loaf, then 1/4 and so on, continually multiplying the denominator by 2. You would always be able to divide the remaining pieces of the loaf in half (although the pieces would eventually become quite minute). In priciple, his claim is correct.

Joe...state: OH
country: USA
comments: The bread could be halved repeatedly, but not forever...
Eventually the bread would lose the characteristics that make it bread, therefore, after so many halvings, the bread would cease to be bread.

Joy Bose...city: Manchester
country: UK
comments: Well, on first thought, it seems that there should be enough bread for everyone. It an infinite series of geometrically progressing numbers 1, 1/2, 1/4 etc.
However, another factor should be taken into consideration here: if we get small enough, we would reach atomic sizes and the bread would no longer be capable of being called 'bread' in the conventional sense of the term.

Dr. Barnette. This is a very interesting problem that could have many different answers, depending on which particular "what if" scenario one chooses to play out. Unfortunately, Gilbert's claim does not stand true when tested with my "what if". Theoretically, many people would receive bread, but no one would end up with any bread to eat. What if there was no limit to the number of times any particular person received bread from someone else? This means that as long as there were at least two customers who took turns "halving" the bread, they could continue to do this until there was only a single molecule of bread left. At this point, the remaining portion could still be divided! Depending on who the customers are, each person who continued to share their portion would end up with a single atom of some various element... That is unless Leibniz happened to be one of the customers, in which case he would end up with a single tasty Monad!

Given that this scenario requires only two persons for it to be possible, and considering this bread is a "favorite with the patrons", I see no problem with meeting the minimum number of participants. In fact, the only way I see for someone to end up with an actual piece of edible bread is for that person to disobey the rules and not divide his or her bread with other patrons. A very interesting predicament!

Troy Williamson, a loyal Zeno's patron, submitted the following, more technically complex, response. In it Troy makes a distinction not represented in the above comments. I wanted to include it in our replies..RB

Troy Williamson...city: Abilene
state: TX
country: USA

comments#1: If you construct a chart showing how the bread is distributed, assuming that each person takes the maximum allowable portion, you will find that one person receives one-fourth of the loaf (1@1/4), one receives one-eighth of the loaf (1@1/8), two receive one-sixteenth (2@1/16), 3@1/32, 5@1/64, 8@128, and so forth. There are two sequences involved in this pattern: the number of people receiving each portion is the Fibonnaci sequence, while the portions received are the inverses of the powers of 2 (beginning with 2 squared).

Because the Fibonnaci numbers are involved, this is not a geometric sequence. However, the further down the sequence you go, the closer it approximates a geometric sequence, since the ratio between consecutive Fibonnaci numbers approximates phi (the golden mean). Thus, the series generated by this problem is an approximation of the geometric series where the first term is 1/4 and the ratio between consecutive terms is phi/2 (which is approximately 0.809016994). This series does converge, since the ratio is less than 1.

The question remains as to what the sum of the infinite series will be. In truth, the series does not approximate the geometric series mentioned above until about the 15th term. Yet it does appear (I haven't quite proven this yet) that the series will have a sum of 1 (i.e., "1" entire loaf of soda bread). Thus, everyone who wants some bread will get some.
comments#2: This is to append the response which I sent yesterday. As it turns out, there is a proven formula for calculating the sum of the Fibonacci numbers divided by the powers of 2. The infinite series 1/2 + 1/4 + 2/8 + 3/16 + 5/32 + 8/64 + ... = 2.

However, the series for which we are seeking the sum is 1/4 + 1/8 + 2/16 + 3/32 + 5/64 + 8/128 + ... which is equal to 1/2 of the series shown in the preceding paragraph. Thus, the sum will be half of that preceding series, meaning that the sum will be 1.

Thus, the summation formula proves that the sum of the series in question will be 1 (i.e., one whole loaf of soda bread).

Sorry I couldn't supply this information with my original submission. (It was a "work in progress," I suppose!)