# The unit fraction problem

Hey Guys !

In university this year I was given a problem to solve ( or get the best possible solution ). It involves unit fractions and trying to get the largest set of unit fractions to add up to 1. So for example a set of unit fractions between 1/2 - 1/1000 what is the largest set of them you can pair together so your set of numbers is as large as possible ? It looks easy ! But is quite a task. For example the best solution I produced was 304. But someone in my class was able to go up to 427.

Here’s my set of numbers:

``````55,62,63,64,66,68,69,70,78,84,
88,90,91,92,93,95,99,105,108,110,
114,115,119,124,126,128,130,132,133,140,
143,144,152,153,154,155,161,165,168,170,
171,174,175,176,180,182,184,186,187,195,
198,203,204,207,208,209,210,216,217,221,
224,228,230,231,232,234,238,243,247,248,
250,252,255,256,260,261,264,266,270,272,
273,276,279,280,285,286,288,290,297,299,
300,304,306,308,310,315,319,320,322,323,
324,325,330,336,340,341,345,348,350,351,
352,360,364,368,370,372,374,375,377,380,
384,390,391,396,399,400,403,405,406,407,
414,416,418,420,425,429,432,434,435,437,
440,442,444,448,450,455,456,459,460,462,
464,468,475,476,480,481,483,486,493,494,
495,500,504,506,510,518,520,522,525,527,
528,532,540,544,546,550,551,552,555,558,
560,561,567,570,572,575,576,580,585,589,
592,594,595,598,600,608,609,612,616,620,
621,624,627,629,630,638,640,644,646,648,
651,660,663,665,666,667,672,680,682,684,
690,693,696,697,700,702,703,704,713,714,
720,725,728,736,738,740,744,748,750,754,
756,759,760,768,770,775,777,779,780,782,
783,792,798,806,810,812,814,816,819,820,
825,828,832,837,840,850,851,858,860,861,
864,868,870,874,875,884,888,896,897,899,
902,903,912,918,924,925,928,930,936,943,
945,946,952,957,960,962,966,984,986,989,
990,992,999,1000
``````

Plug these into an array add the answers over `1/n[i]` and you can see that it does add up to one !

How did I do it ?

Take a unit fraction and it two factors ( I always went with the smallest one’s ), add the factors together and then multiply the added total by each factor and by that you can find the two unit fractions which just made up the bigger one ! So for e.g.

1/2 = ( 2 + 1 ) = 3(2) = 6 , 3(1) = 3
1/3 + 1/6 = 1/2

Why am I posting this ?

I am curious if anyone can use this pattern to cook up some cool visuals give it a go ! I will definitely have something myself but just I am busy currently.

Have fun !