Can you finish eating a cake if you are required to take a bite only half of it on hand at a time?

Question

You have one piece of cake . you are required to finish eating it . At a time you should take only half of what you have onhand . when can you finish eating the whole cake? (Say it is not very big .say it is just two cubic inches only in size).
What does theresult indicate or teach us .?

Answer 1

You will never finish the cake until you get it down to two atoms. Every time you have to leave half of it behind. This will quickly result in a small piece, but the pieces you are taking off of the cake are always half as small. Which creates an infinite cycle. Eventually you will reach an end, but not for a very, very long time.

Another example of this is the infinite race. If you run half the race each day, how long will it take to finish the race? Since you are only running half of the race each day the distance drops quickly, but it never quite reaches zero. Pretty soon you will be taking baby steps each day, but you will never finish the race. As you get closer to the finish line you will be moving the same distances that are present inside atomic particles, but that still is a positive distance. You will just barely reach the finish line, but you will never cross it.

Radioactive materials are measured in Half-Lives. After one year half of the radioactive sample will have emitted enough radiation that it will no longer be radioactive. The problem is that for each passing year only one half of the remaining material losses its radioactivity. That's what makes radioactive material so dangerous; it will be around for a very long, long time. Like your piece of cake the radioactive part will be their in increasingly smaller bits, but it will almost never go away.

As you continue your education in mathematics you will be exposed to this concept even more. Some formulas (number and variable functions) will never reach a firm answer. They will approach a number but not quite reach it. This number is known as a limit. If you had a space craft that went twice as fast each day then you would soon be rocketing off at speeds that would shame Star Trek's Enterprise. Except that the speed of light is a limit. You can go faster and faster, but you can never really reach the speed of light. What actually happens as you approach the speed of light was discovered in Einstein’s theory of Special Relativity. As you get faster and faster your mass increases, and your time slows down (your dimension along the axis of travel also decreases). The theory has been proven to be correct, experiments with atomic clocks on airplanes and the space shuttle have shown that as you move faster your relative time does slow down. As you approach the speed of light you will weigh more and more, and your time will pass slower and slower. Once you get to 999.99% of the speed of light your relative time will almost stop, and you will almost weigh an infinite mass. To actually reach the speed of light you would be frozen in time and have an infinite mass. This can't happen in the normal universe so you can never reach light speed. If you did the laws of physics would break down.

This is what makes a black hole so interesting. At some point objects falling into it would reach a speed approaching the speed of light. At this point to get any information from the black hole would require a signal to be sent at a speed faster than the speed of light; this can’t happen in the normal universe. So each black hole has a point where you can’t get any more information, because the information signal can’t go fast enough to escape the black hole. This limit is called the event horizon. You can never see inside a black hole because the event horizon shields it. Since the gravity in a black hole is all concentrated at one point, objects can pass the event horizon and continue to fall faster and faster. If this happens then the laws of physics break down and ANYTHING can happen. If you could somehow see past the black hole’s event horizon then you wouldn’t understand what you saw. Things could happen like time flowing backward, or electricity flowing backward. Even stranger stuff could happen. That’s the point we don’t know what happens inside the event horizon, and out understanding of physics breaks down so we never will.

Answer 2

This is a rephrasing of Zeno's paradox. See summary at link below:

Several of the comments given above make sense. But if you assume that the rate at which you eat the cake remains constant (ie you can eat 1/2 the cake in 1 min, 1/4 in 30 sec, 1/8 in 15 sec, etc) then you can use the formula for the sum of a convergent series and the entire cake will be eaten within a finite amount of time.

In my example, the entire cake will be eaten in 2 mins.
Sum = first term / (1 - ratio) = 1min / (1 - 0.5) = 2 mins.

Answer 3

No, you could never finish it, under those conditions.... well, until it stops *being* cake. At some point it will be the smallest possible collection of molecules which make it cake (or the ingredients in cake, at least) and taking half of those molecules and the atoms that comprise them will change it to some other material(s).

It wouldn't matter if you started with a cubic centimeter, or a crumb.

Hmm. I'm not sure what that teaches us... other than how to save snacks for later. ;-)

It does nicely demonstrate the concept of infinity.

Answer 4

Consider this...
0.5 + 0.25 + 0.125 + 0.0625 + 0.03125 + 0.015625 = 0.984375

So first you eat half then quarter then 1/8th etc. Theoretically, if you keep doing that you finish the cake, but then if you think about it, you can't, because after each bite you must always have 'the other half' in your hand otherwise you're not following the rules(or whatever).

Sum of an infinite series is given by A/(1-R) where A is the first term( in this case 0.5) and R is the factor by which a term is multiplied in order to get the next term( in this case it's half as you're eating 'half' of what is left). So if you calculate you get 1 => one cake.

Answer 5

that there are too many ppl in the world without bloody common sense. rubbish infinite times. probably about 5-7 bites and the cake is too small to be divided by the mouth/teeth, then u would have to eat the whole damn thing.
if it was a cheese cake though i would have taken 2 bites 2 finish the whole damn thing, and called for 3 more rounds all in the space of 2 seconds...lol

Answer 6

Cake is not an infinite series, it's quantized.
The real question is how well accurately you can eat your defined portions.
Since I doubt you can eat to that degree of precision, I'm going with never.

Keeping track of all the molecules would be a trick too, since you have to decide whether or not the aroma counts as being part of the mass of the cake.

Answer 7

It teaches us that Zeno is alive and well. :-)

Practically, of course, you will quickly reach a point at which dividing the remaining cake accurately in half by even the most delicate of means, much less one's teeth, is impossible. In fact, if the cake starts at 2 in³ in size, I'd say it would become too small to "bite only half of it" accurately within 10 bites.

Answer 8

Such Qs were asked before people knew about zero. Askers held sway and perplexed Kings and Queens with so-called contradictions between life and mathematics. This also gave an eerie sense of ideas such as from where I come or where will I go. Zero is supposed to have solved this puzzle and made math a lot credible.

Sorry, I don't know enough of math to remember whatever I read in Google!

Answer 9

this is a typical example for limit.theoretically you will never be able to eat the entire cake.but for all practical purposes after about 10 to 15 bites you would have finished the cake up

Answer 10

Yes, you'll finish it but it will take an infinit amount of time and an infinite number of bites.

What it teaches us is that

1/2 + 1/4 + 1/8 + 1/16 + ***** to infinity = 1


Doug