Wheat and chessboard problem.

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When the creator of the game of chess (in some tellings an ancient Indian mathematician, in others a legendary dravida vellalar named Sessa or Sissa) showed his invention to the ruler of the country, the ruler was so pleased that he gave the inventor the right to name his prize for the invention. The man, who was very wise, asked the king this: that for the first square of the chess board, he would receive one grain of wheat (in some tellings, rice), two for the second one, four on the third one, and so forth, doubling the amount each time. The ruler, arithmetically unaware, quickly accepted the inventor’s offer, even getting offended by his perceived notion that the inventor was asking for such a low price, and ordered the treasurer to count and hand over the wheat to the inventor. However, when the treasurer took more than a week to calculate the amount of wheat, the ruler asked him for a reason for his tardiness. The treasurer then gave him the result of the calculation, and explained that it would take more than all the assets of the kingdom to give the inventor the reward. The story ends with the inventor becoming the new king.

 

If a chessboard were to have wheat placed upon each square such that one grain were placed on the first square, two on the second, four on the third, and so on (doubling the number of grains on each subsequent square), how many grains of wheat would be on the chessboard at the finish?   The total number of grains equals 18,446,744,073,709,551,615 which is a much higher number than most people intuitively expect.

 

Another version places two merchants together. One merchant offers the other a deal; that for the next month, the merchant was going to give $10,000 (or, in some variants, even $100,000) to the other one, and in return, he would receive 1 cent the first day, 2 cents the second, 4 cents on the third, and so on, each time doubling the amount. The second merchant agreed, and for the first three weeks (or more, depending on the variant), he enjoyed the fortunes that the first merchant was “unwittingly” giving him, but by the end of the month, the second merchant was broke, while the first merchant was incredibly rich.  

 

Yet another variant is about a man buying a horse and displeased with the high price. The owner offers him to buy it by paying instead one cent for the first nail in its horseshoes, two for the second nail, and so on. With each horseshoe having six nails, the results are similar to the above story.  

 

Some books about popular mathematics also mention this problem: Fold a Post-it note in half seven times, and then try to rip the note. Folding the paper would soon become difficult as the note becomes increasingly smaller and increasingly harder to manipulate. However, even if you were to start with a large piece of paper, you would be unable to rip it because the paper would be 128 sheets thick.

 

Many popular internet scams are based on the above principle. These include pyramid schemes, matrixes and chain letters to name a few.

 

Common methods used in these schemes include emotionally manipulative stories, get-rich-quickly pyramid schemes, and the exploitation of superstition to threaten the recipient with bad luck or even physical violence or death if he or she “breaks the chain” and refuses to adhere to the conditions set out in the scheme. They are commonly marketed through email messages, postings on social network sites, and text messages.

 

The flaw in these scams is obviouse, once the chain is broken the doubling effect stops and you loose.

 

In order for the scheme to work you are dependant not only on your own marketing abilities, but also on your downline continuing the chain by recruiting new prospects who must then in turn continue building the chain. As soon as one of these legs stop and so break the chain, so has your doubling principle.

 

Most of us have experienced joining programs that suddenly disappear overnight leaving us out of pocket. Unless you are the originator of the scheme, or one of the first to join, you are unlikely to recoup your investment by joining such programs. The chance of turning a profit is minimal and not worth the risk.

 

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