Things I ponder while I am here at work

Stu · **Posted:** Wed Jul 14, 2004 5:18 pm

Most of you have already heard this (I know I first heard about it years ago in junior high). But the story goes something like this. You have performed some great task for the king (or another great and rich fellar) and he has agreed to pay you. He tells you that he will give you a million dollars.

You think about it, and come up with another plan which you present to him. It is the month of July (the first to be exact) and tell the king that you would be happy if he would just give you money according to this plan. The first day he gives you a penny, the second day he gives you two pennies, the third day he gives you four pennies, and every day he gives you double the previous days earnings.

The king happily agrees. So, how much do you end up making?

Kinda nifty if you ask me.

Code:

(2^31) + (2^30) + (2^29) + (2^28) + (2^26) + (2^25) + (2^24) + (2^23) + (2^22) + (2^21) + (2^20) + (2^19) + (2^18) + (2^17) + (2^16) + (2^15) + (2^14) + (2^13) + (2^12) + (2^11) + (2^10) + (2^9) + (2^8) + (2^7) + (2^6) + (2^5) + (2^4) + (2^3) + (2^2) + (2^1) = 4,160,749,566

over 4 billion pennies (or 41.6 million dollars). On the last day alone you are given over 2 billion.

Another page worth checking out is this one.

As I said, this is just stuff that I do when I am bored at work.

Tom · **Posted:** Wed Jul 14, 2004 5:45 pm

Stu wrote:

Code:

(2^31) + (2^30) + (2^29) + (2^28) + (2^26) + (2^25) + (2^24) + (2^23) + (2^22) + (2^21) + (2^20) + (2^19) + (2^18) + (2^17) + (2^16) + (2^15) + (2^14) + (2^13) + (2^12) + (2^11) + (2^10) + (2^9) + (2^8) + (2^7) + (2^6) + (2^5) + (2^4) + (2^3) + (2^2) + (2^1) = 4,160,749,566

over 4 billion pennies (or 41.6 million dollars). On the last day alone you are given over 2 billion.

Umm... hang on. I'm checking that math...

Hmm. Check this...

Code:

2^1=    2
2^2=    4
2^3=    8
2^4=    16
2^5=    32
2^6=    64
2^7=    128
2^8=    256
2^9=    512
2^10=   1024
2^11=   2048
2^12=   4096
2^13=   8192
2^14=   16384
2^15=   32768
2^16=   65536
2^17=   131072
2^18=   262144
2^19=   524288
2^20=   1048576
2^21=   2097152
2^22=   4194304
2^23=   8388608
2^24=   16777216
2^25=   33554432
2^26=   67108864
2^27=   134217728
2^28=   268435456
2^29=   536870912
2^30=   1073741824
2^31=   2147483648

Total=  4294967294

Yeah, so you were close. But really that calculation is if you are given two pennies on the first day. If it was only one penny and the month was 31 days long, you'd only get 2,147,483,646 pennies.

I've also seen it done with geometry.

Math = fun.

InterruptorJones · **Posted:** Wed Jul 14, 2004 6:02 pm

This is one of many variation on the old chess board fable. It goes like this (this is from memory, and I'm embellishing plenty):

Ages ago, in Persia, the king became bored with all of the games he had to play in the palace, and charged his subjects the task of inventing a new game for him, promising to reward the inventor of a game suited for a king with the prize of his choice. Many subjects, some of them great scholars and lords, came and went, and the king found every one of the new games trite and boring (in some versions of the story he has these beheaded), until one day when an old peasant approached his throne proferring a simple square board with eight smaller squares to a side and a sachel of small carved figurines. The game was chess, and the king was immediately taken by it. He and the peasant played for hours until the king became quite skilled at the game.

Finally, the as the sun was setting, the king declared, "Peasant, your game is delightful; name your reward!"

The peasant, having already put much thought into the prize he would chose, replied, "your majesty, I am but a simple man and wish only to provide for my family..." He then cleared the chess board and described to the king the reward he desired. "Tomorrow," he explained, "you will place a single grain of rice on the first square of this chess board, and that rice will be mine. The next day, you will place two grains of rice on the next square, and that rice will be mine. And the next day you will place four grains of rice, and every day thereafter twice as many grains of rice than the previous day, until sixty-four days have passed. I will take this rice to my family and be happy."

The king guffawed. "Such a paltry sum for so a kingly a game as this? A few grains of rice are nothing to a king! Your wish is granted, now return to the countryside and your thatch-roofed cottage."

And the peasant went home happy.

Or something.

Anyway, the way it works out is that by the end of the 64 days, the king is starving and the peasant has 18,446,744,073,709,551,616 grains of rice, which, assuming a tenth of a gram per grain (arbitrary guess), is more than two trillion tons of rice.

Stu · **Posted:** Wed Jul 14, 2004 6:08 pm

Tom wrote:

Umm... hang on. I'm checking that math...

Google does my math.

Stu · **Posted:** Wed Jul 14, 2004 6:10 pm

InterruptorJones wrote:

This is one of many variation on the old chess board fable. It goes like this (this is from memory, and I'm embellishing plenty):

Ages ago, in Persia, the king became bored with all of the games he had to play in the palace, and charged his subjects the task of inventing a new game for him, promising to reward the inventor of a game suited for a king with the prize of his choice. Many subjects, some of them great scholars and lords, came and went, and the king found every one of the new games trite and boring (in some versions of the story he has these beheaded), until one day when an old peasant approached his throne proferring a simple square board with eight smaller squares to a side and a sachel of small carved figurines. The game was chess, and the king was immediately taken by it. He and the peasant played for hours until the king became quite skilled at the game.

Finally, the as the sun was setting, the king declared, "Peasant, your game is delightful; name your reward!"

The peasant, having already put much thought into prize he would chose, replied, "your majesty, I am but a simple man and wish only to provide for my family..." He then cleared the chess board and described to the king the reward he desired. "Tomorrow," he explained, "you will place a single grain of rice on the first square of this chess board, and that rice will be mine. The next day, you will place two grains of rice on the next square, and that rice will be mine. And the next day you will place four grains of rice, and every day thereafter twice as many grains of rice than the previous day, until sixty-four days have passed. I will take this rice to my family and be happy."

The king guffawed. "Such a paltry sum for so a kingly a game as this? A few grains of rice are nothing to a king! Your wish is granted, now return to the countryside and your thatch-roofed cottage."

And the peasant went home happy.

Or something.

Anyway, the way it works out is that by the end of the 64 days, the king is starving and the peasant has 18,446,744,073,709,551,616 grains of rice, which, assuming a tenth of a gram per grain (arbitrary guess), is more than two trillion tons of rice.

I remember now, I do recall hearing this version of it. Now, if I was the king, once I realized what was going on I would have had that clever peasant "disposed" of. Nowhere does it say that the king has to play fair, sheesh.

Tom · **Posted:** Wed Jul 14, 2004 6:21 pm

Okay. I'm done. See my previous post for the math.

Oh, one more thing, Stu's (er, Google's) percent error is...

Code:

((4294967294-4160749566)/4160749566)*100=3.125000001

InterruptorJones · **Posted:** Wed Jul 14, 2004 6:40 pm

Oh, hey, my math was wrong. Or just lazy, I guess. The number of grains of rice I posted was just for the last day. The actual sum (according to Excel) would be on the order of 3.68935E+19, or more than 4 trillion tons.

Tom · **Posted:** Wed Jul 14, 2004 6:52 pm

Oh yeah, Google isn't wrong either. It gave Stu the answer for the question he asked. Stu just asked the wrong question.

He forgot the

Code:

(2^27)

Throws the answer off by those 134,217,728 pennies.

Gemini · **Joined:** Mon May 10, 2004 6:40 pm **Posts:** 273

InterruptorJones wrote:

Anyway, the way it works out is that by the end of the 64 days, the king is starving and the peasant has 18,446,744,073,709,551,616 grains of rice, which, assuming a tenth of a gram per grain (arbitrary guess), is more than two trillion tons of rice.

That peasant's a jerk. I woulda asked for like, half the kingdom.

Stu · **Posted:** Wed Jul 14, 2004 8:28 pm

Tom wrote:

Oh yeah, Google isn't wrong either. It gave Stu the answer for the question he asked. Stu just asked the wrong question.

Silly me.

as for the 2^27, I have no idea how I missed that. Strange. (?)

Strong Lad · **Posted:** Wed Jul 14, 2004 9:23 pm

Heard it, but slightly different. In the story that I read it was in a book titled THE KNOWLEDGE: Murderous Maths. (The "S" is meant to be there. The British abbreviate "mathematics" to Maths, not Math. I wouldn't point this out, but ever since someone ranted to Movie-Mistakes about the word "Maths" being used, I'm being cautious.)

Anyhoo...

The story goes that a general is in charge of an army, and he arrives at the barracks to find that they had all wagered their clothes against a mathemagician (sic). He is angry at them, and demads to play against the mathemagician for their clothes. He fails, and is given the option of paying 1 pound for the first, 2 pounds for the next, 3 pounds for the next etc etc etc. The other option, which he chooses, is 1 penny for the first, 2 pence for the next, 4 pence for the next and so on. The story ends with the revelation described by Stu. But it IS pretty amazing, innit?

furrykef · **Posted:** Thu Jul 15, 2004 12:16 am

Strong Lad wrote:

I wouldn't point this out, but ever since someone ranted to Movie-Mistakes about the word "Maths" being used, I'm being cautious.

On the behalf of all my peoples, I am embarrassed.

(And yeah, that "s" is meant to be there, though it's improper grammar.

)

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