# Avery's Bonus

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.

# The first row: numbers

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

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

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

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

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

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.
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

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.