Before the school year began, I ran into our esteemed headmaster Larry Weiss while rummaging for books on the fourteenth floor of Saint Ann's. After a brief conversation about summer and rhinoceroses, Larry noted that he had heard much about my students and their mathematical accomplishments--apparently he had found out that every single Nobel Prize for Mathematics winner in the past five years was a student of mine. As a token of appreciation, he insisted on giving me a bonus. Blushing, I assured him that my students' success was due to their hard work, curiosity, and deep understanding of mathematics, and that they should be the ones getting a bonus. He insisted, however, and told me that I could choose between two options. Option one: $1,000,000. Option two: a bunch of salt (apparently he had received a generous parting gift from his old students). Being a mathematician, my response was obviously...How much salt? He then explained that he would use a chessboard and would place 1 grain of salt on the first square of the board on the first day of school, 2 grains of salt on the second square of the board on the second day of school, 4 grains of salt on the third square on the third day of school, etc. I told him that his offer was intriguing, but $1,000,000 was a lot of money. He assured me that the choice was mine. So now I leave it to you, my hard working, curious and telented students to help me figure out which option I should choose...and maybe we'll learn a little about numbers and measuring along the way.

## Square |
## Date |
## Day bonus |
## Total bonus |
## More on today's bonus... |

## The first row: numbers |
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1 | 9/12 | 1 | 1 | 1x1=1, 11x11=121, and 111x111=12321. What about 1111x1111? Is your answer a conjecture or a proof? |

2 | 9/13 | 2 | 3 | 2 x 2 = 2 + 2. Can you find another number this works for? |

3 | 9/14 | 4 | 7 | Also known as Quatro, Quatre, Vier, Tchetuire, and Shi (which is considered bad luck in Japan because it's the same pronunciation as the word for 'death'). |

4 | 9/18 | 8 | 15 | A shape with eight equal sides is an octogon. An octave consists of eight notes. Can you think of other oct- words? |

5 | 9/19 | 16 | 31 | 2 multiplied by itself 4 times (2x2x2x2) = 4 multiplied by itself 2 times (4x4) = 16. Can you find another two numbers this works for? |

6 | 9/20 | 32 | 63 | My total bonus is always going to be odd. Can you 'prove' why? |

7 | 9/21 | 64 | 127 | 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64. Explain how the numbers on the left are related. |

8 | 9/25 | 128 | 255 | Look back on the first row. Notice anything about the total bonuses compared to the day bonuses? Make a conjecture based on any pattern you see. |

## The second row: weights |
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9 | 9/26 | 256 | 511 | Today's bonus weighs 11 grams, about the same as 11 paperclips. |

10 | 9/27 | 512 | 1,023 | If every grain weighs the same, no scale is needed to know this will weigh 22 grams. |

11 | 9/28 | 1,024 | 2,047 | And today's bonus will weigh how much? |

12 | 9/29 | 2,048 | 4,095 | Nutritionalists recommend eating close to 88 grams of salt per month. |

13 | 10/3 | 4,096 | 8,191 | 176 grams of salt would be consumed in about 40 McDonald's meals. |

14 | 10/6 | 8,192 | 16,383 | $16 worth of quarters (64 quarters) weigh about 352 grams. |

15 | 10/11 | 16,384 | 32,767 | There is about 704 grams of salt in 5 gallons of ocean water. |

16 | 10/12 | 32,768 | 65,535 | This morning, 1408 grams (about 3 pounds) of salt were sitting on my desk. |

## The third row: areas |
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17 | 10/17 | 65,536 | 131,071 | If each grain were 1mm by 1mm, 65,536 grains of salt would cover a 10 inch square. |

18 | 10/18 | 131,072 | 262,143 | This would NOT cover a 20 inch square. Can you figure out why? |

19 | 10/19 | 262,144 | 524,287 | This, however, would cover a 20 inch square. |

20 | 10/20 | 524,288 | 1,048,576 | This would cover a circle with a diameter of about 32 inches. |

21 | 10/24 | 1,048,576 | 2,097,151 | This would cover close to the entire side of our chalkboard. |

22 | 10/25 | 2,097,152 | 4,194,303 | So today's salt would cover both sides of the chalkboard. |

23 | 10/26 | 4,194,304 | 8,388,607 | This would almost cover a 10 ft by 5 ft rectangle. |

24 | 10/27 | 8,388,608 | 16,777,215 | So today's salt would cover about one-third of the classroom floor. |

## The fourth row: times |
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25 | 10/31 | 16,777,216 | 33,554,431 | If I could count one grain per second, this would take almost 32 years to count. |

26 | 11/1 | 33,554,432 | 67,108,863 | At 1 grain a second, this would take almost 64 years; at ten grains per second, 6.4 years. |

27 | 11/2 | 67,108,864 | 134,217,727 | The oldest person in the world is 117, 11 years less than the time to count this salt. |

28 | 11/3 | 134,217,728 | 268,435,455 | I would have had to start counting before the Revolutionary War to count this salt. |

29 | 11/7 | 268,435,456 | 536,870,911 | If I could count one grain per second, this would take almost 512 years to count. |

30 | 11/8 | 536,870,912 | 1,073,741,823 | ...1024 years to count. |

31 | 11/9 | 1,073,741,824 | 2,147,483,647 | ...2048 years to count. |

32 | 11/10 | 2,147,483,648 | 4,294,967,295 | ...4096 years to count. How long is this going to take 10 squares from now? (hint: look at squares 11, 12, and 13) |

## The fifth row: distances |
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33 | 11/14 | 4,294,967,296 | 8,589,934,591 | If every grain of salt were 1 millimeter long, today's salt would be 2,668 miles long...about the distance from New York to Las Vegas. |

34 | 11/15 | 8,589,934,592 | 17,179,869,183 | 5,336 miles of salt will get you from New York to Buenos Aires, Argentina. |

35 | 11/16 | 17,179,869,184 | 34,359,738,367 | 10,672 miles of salt will get you from New York to Hong Kong. |

36 | 11/17 | 34,359,738,368 | 68,719,476,735 | 21,344 miles is close to the distance that the Arctic Tern travels in one year, migrating from the north pole to the south pole and back. |

37 | 11/21 | 68,719,476,736 | 137,438,953,471 | 42,688, or twice around the world. |

38 | 11/22 | 137,438,953,472 | 274,877,906,943 | Driving at 60 mph, it would take me two months of straight driving to travel 85,376 miles. |

39 | 11/28 | 274,877,906,944 | 549,755,813,887 | 170,752 miles |

40 | 11/29 | 549,755,813,888 | 1,099,511,627,775 | 341,504 miles of salt will get you from New York to Cape Canaveral, Florida to the moon (and almost halfway back). |

## The sixth row: dollars |
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41 | 11/30 | 1,099,511,627,776 | 2,199,023,255,551 | This weighs 50,332 tons and, at the bulk rate of $40/ton (what New Hampshire pays for road salt), would cost $2,013,266. |

42 | 12/1 | 2,199,023,255,552 | 4,398,046,511,103 | Could sell this for $4,026,532. Click here for more information than you could ever want about road salt use. |

43 | 12/5 | 4,398,046,511,104 | 8,796,093,022,207 | 201,328 tons of salt worth $8,053,064. Remember that I can sell my total salt for twice this price (minus the price of one grain). |

44 | 12/6 | 8,796,093,022,208 | 17,592,186,044,415 | 402,656 tons of salt worth $16,106,128. |

45 | 12/7 | 17,592,186,044,416 | 35,184,372,088,831 | 805,312 tons of salt worth $32,212,256. |

46 | 12/8 | 35,184,372,088,832 | 70,368,744,177,663 | 1,610,624 tons of salt worth $64,424,512. |

47 | 1/3 | 70,368,744,177,664 | 140,737,488,355,327 | 3,221,248 tons of salt worth $128,849,012. |

48 | 1/4 | 140,737,488,355,328 |
281,474,976,710,655 | 6,442,496 tons of salt worth $257,698,024. |

## The seventh row: volumes |
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49 | 1/5 | 281,474,976,710,656 | 562,949,953,421,311 | This would fill a box that is 215 feet on each side (a little less than a block). This is also about how much salt is used on US roads in a very snowy year. |

50 | 1/9 | 562,949,953,421,312 | 1,125,899,906,842,623 | This would fill all of the Saint Ann's buildings. |

51 | 1/10 | 1,125,899,906,842,624 | 2,251,799,813,685,247 | This would almost fill the Empire State Building. |

52 | 1/11 | 2,251,799,813,685,248 |
4,503,599,627,370,495 | This woud fill the tallest building in the world, Taipei 101, which opened on New Year's Eve and is 1,671 feet tall. |

53 | 1/12 | 4,503,599,627,370,496 | 9,007,199,254,740,991 | This would fill a box that is 540 feet on each side. |

54 | 1/17 | 9,007,199,254,740,992 | 18,014,398,509,481,983 | This would fill a box that is 682 feet on each side (more than three city blocks). Why wouldn't it be 1,080 ft? |

55 | 1/19 | 18,014,398,509,481,984 | 36,028,797,018,963,967 | This would more than fill the Boeing Everett Factory in Seattle, the largest building (by volume) in the world. |

56 | 1/23 | 36,028,797,018,963,968 | 72,057,594,037,927,935 | This would fill a fifty-story building that took up five square blocks. |

## The eighth row: numbers, weights, areas, times, heights, dollars, and volumes |
||||

57 | 1/24 | 72,057,594,037,927,936 | 144,115,188,075,855,871 | Could be thought of as seventy-two quadrillion, fifty-seven trillion, five hundred ninety-four billion, thirty-seven million, nine hundred twenty-seven thousand, nine hundred thirty-six, 3,413,003,560 tons, 124 square miles, 2,284,931,318 years, 44,774,513,100 miles, $136,520,142,400, or 1,364 cubic feet. |

58 | 1/25 | 144,115,188,075,855,872 | 288,230,376,151,711,743 | After quadrillion comes sextillion, septillion, octillion, nonillion and decillion. |

59 | 1/26 | 288,230,376,151,711,744 | 576,460,752,303,423,487 | Although 6,597,069,764 tons is ten times more than the combined weight of every human being, this is about how much carbon dioxide is released into the air each year by humans. |

60 | 1/30 | 576,460,752,303,423,488 | 1,152,921,504,606,846,975 | 472 square miles would more than cover the 321 square miles of New York City. |

61 | 1/31 | 1,152,921,504,606,846,976 | 2,305,843,009,213,693,951 | The 36,558,901,088 years it would take to count this salt at one grain per second is more than twice the age of the universe. |

62 | 2/1 | 2,305,843,009,213,693,952 | 4,611,686,018,427,387,903 | Stacked, this would extend 1,432,784,419,200 miles which is more than 15,000 trips to the sun but only one-seventeenth of the way to the next nearest star. |

63 | 2/2 | 4,611,686,018,427,387,904 | 9,223,372,036,854,775,807 | $8,737,289,113,600 would completely pay off the national debt with a trillion dollars to spare. |

64 | 2/6 | 9,223,372,036,854,775,808 | 18,446,744,073,709,551,615 | The 2,097 cubic meter box that would be needed to hold this salt is more than a mile on each side. |