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Math Series Part II: Doubling
We can learn a lot about the world through mathematics. Indeed, many stories throughout history have even shaped our understanding of mathematics. Here, we look at how complex a number can become given a small choice about what to do with it over time.
Folding to the Checkmate: The Exponential Growth Involved with Doubling
The creation of the chessboard brings with it an interesting tale of the reward requested by its inventor. In recognition of the incredible game invented, the king offers the creator a prize of his choosing. The response was one seen as a meager and simple request by the then ruler and was swiftly granted to the sly innovator: he asked that he be given a single piece of rice for the first square of his chessboard and that this amount be continuously doubled for the remaining squares of the chessboard, which totaled 64. Not prone to the complexities of the mathematics involved, the ruler only found out the true cost of his decision after his treasuries reported back to him the enormity of the reward.It was incalculable.
By doubling every grain of rice 63 times, starting with a single grain on the first square of the chessboard, the ruler would have had to surrender more than one billion grains of rice for the first 29 squares alone! Indeed, we are able to calculate just how many rice gains were due for an individual square, x, by using the equation x = 2^{n}, where n is the square number in the sequence from 0 to 63. The table below shows just what an incredible amount of rice would have been granted if the creator had changed his request for 1 grain for the first square to two; more than eighteen quintillion for the final square!
n = 1: 2^1= 2
 n = 2: 2^2= 4
 n = 3: 2^3= 8
 n = 4: 2^4= 16


n = 5: 2^5= 32
 n = 6: 2^6= 64
 n = 7: 2^7= 128
 n = 8: 2^8= 256

n = 9: 2^9= 512
 n = 10: 2^10= 1024
 n = 11: 2^11= 2048
 n = 12: 2^12= 4096

n = 13: 2^13= 8192
 n = 14: 2^14= 16384
 n = 15: 2^15= 32768
 n = 16: 2^16= 65536

n = 17: 2^17= 131072
 n = 18: 2^18= 262144
 n = 19: 2^19= 524288
 n = 20: 2^20= 1048576

n = 21: 2^21= 2097152
 n = 22: 2^22= 4194304
 n = 23: 2^23= 8388608
 n = 24: 2^24= 16777216

n = 25: 2^25= 33554432
 n = 26: 2^26= 67108864
 n = 27: 2^27= 134217728
 n = 28: 2^28= 268435456

n = 29: 2^29= 536870912
 n = 30: 2^30= 1073741824
 n = 31: 2^31= 2147483648
 n = 32: 2^32= 4294967296

n = 33: 2^33= 8589934592
 n = 34: 2^34= 17179869184
 n = 35: 2^35= 34359738368
 n = 36: 2^36= 68719476736

n = 37: 2^37= 137438953472
 n = 38: 2^38= 274877906944
 n = 39: 2^39= 549755813888
 n = 40: 2^40= 1099511627776

n = 41: 2^41= 2199023255552
 n = 42: 2^42= 4398046511104
 n = 43: 2^43= 8796093022208
 n = 44: 2^44= 17592186044416

n = 45: 2^45= 35184372088832
 n = 46: 2^46= 70368744177664
 n = 47: 2^47= 140737488355328
 n = 48: 2^48= 281474976710656

n = 49: 2^49= 562949953421312
 n = 50: 2^50= 1125899906842624
 n = 51: 2^51= 2251799813685248
 n = 52: 2^52= 4503599627370496

n = 53: 2^53= 9007199254740992
 n = 54: 2^54= 18014398509481984
 n = 55: 2^55= 36028797018963968
 n = 56: 2^56= 72057594037927936

n = 57: 2^57= 144115188075855872
 n = 58: 2^58= 288230376151711744
 n = 59: 2^59= 576460752303423488
 n = 60: 2^60= 1152921504606846976

n = 61: 2^61= 2305843009213693952
 n = 62: 2^62= 4611686018427387904
 n = 63: 2^63= 9223372036854775808
 n = 64: 2^64= 18446744073709551616

Incredibly, this same form of exponential doubling can occur in all areas of life, including with bacteria and population growth. And, it also happens when we try to fold an infinitely large sheet of paper. For instance, if we had a giant sheet of paper with a thickness of 0.1 millimeters, the same equation of x = 2^{n} applies, where x is the combined thickness of the vertical folds and n is the number of folds. The only difference this time is that the paper’s thickness is not a whole number, as the grains were for the chessboard, and we have to transform the equation to x = (0.1)*2^{n}. Thus, our thickness will have values in millimeters that are one tenth those above in the table, meaning that after just 35 folds, the paper will have reached almost 3.5 thousand kilometers high. With the sun being 149.6 million km from earth, we would only need to fold our paper 51 times in order to reach, and surpass, it because of the exponential growth rate.
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