Can't wait for next year!

Let A A be the sum of digits in 201 5 2015 2015^{2015} .
Let B B be the sum of digits in A A .
Let C C be the sum of digits in B B .
Find A + B + C A+B+C .


The answer is 29517.

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2 solutions

Arulx Z
Aug 28, 2015

With current computation speed, this problem can be solved well under a second even after expanding the whole thing. Here's my code:

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>>> a = sum(int(x) for x in str(2015 ** 2015))
>>> b = sum(int(x) for x in str(a))
>>> c = sum(int(x) for x in str(b))
>>> a + b + c
29517

Solves in 0.002 sec.

Moderator note:

Simple standard approach.

A variant could be:

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>>> def sumdigit(n):
    return sum(map(int, str(n)))

>>> A=sumdigit(2015**2015)
>>> B=sumdigit(A)
>>> C=sumdigit(B)
>>> A+B+C
29517

Abdelhamid Saadi - 5 years, 9 months ago

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Yes that would make the code shorter and more readable. Thanks for posting the code!

Arulx Z - 5 years, 9 months ago

exactly what i did.

Aareyan Manzoor - 5 years, 9 months ago
April Husband
Nov 15, 2014

First, it's a big number, but it's not difficult to solve the initial problem:

2015^2015 = 13,008,484,727,577,632,770,737,373,237,923,736,420,778,606,563,938,581,509,427,661,704,420,083,647,581,495,646,678,499,377,112,942,090,385,715,199,467,773,086,766,900,924,619,108,103,079,812,466,437,693,848,999,949,811,799,512,204,771,242,406,815,142,577,148,488,246,753,963,859,033,887,504,337,237,935,088,014,023,086,066,029,245,019,244,843,354,553,937,209,641,597,306,260,137,143,240,737,786,903,029,642,143,571,514,565,230,861,125,379,473,937,672,071,399,421,824,772,038,156,894,455,234,074,942,040,487,507,066,443,265,280,611,320,726,480,874,516,006,571,976,421,013,353,086,240,591,302,505,984,268,236,061,607,213,191,017,486,003,189,654,910,404,016,969,702,396,575,775,851,422,034,388,257,915,523,677,322,011,737,744,064,822,762,169,393,931,322,031,806,428,327,604,490,381,409,058,200,692,578,911,465,537,205,389,961,160,004,907,508,001,432,049,209,863,231,008,731,822,347,647,136,745,839,173,928,999,212,212,932,157,544,799,251,732,035,662,505,579,765,749,918,879,285,968,891,288,124,663,570,101,587,397,869,608,294,831,077,045,153,963,628,403,101,490,176,213,382,147,759,098,911,092,978,731,831,626,452,835,853,755,475,326,357,732,580,685,367,462,167,731,471,916,339,663,408,447,460,925,492,126,680,875,000,926,517,577,228,224,918,341,018,114,548,512,183,054,601,474,047,984,598,215,118,700,188,856,559,019,103,124,657,837,241,273,506,359,575,950,640,402,603,211,635,337,872,576,195,885,426,245,406,449,102,451,624,682,642,967,104,562,431,902,103,383,948,226,806,085,017,824,702,082,495,206,080,372,090,101,602,849,127,427,326,529,986,924,751,094,267,538,426,016,326,405,951,434,176,215,218,523,695,460,620,960,823,619,640,429,856,332,400,585,398,711,325,939,147,219,894,707,250,186,162,365,102,114,680,216,079,235,865,775,676,838,644,494,940,490,879,597,712,265,085,597,081,915,041,096,698,213,672,511,550,250,230,251,871,201,189,566,224,070,026,088,791,514,440,786,130,153,300,945,081,357,503,035,941,674,486,191,608,197,007,952,772,593,289,952,128,257,664,253,453,262,647,952,910,869,861,026,753,344,801,691,064,946,695,778,961,408,368,489,147,986,496,053,199,009,893,338,416,300,933,263,728,073,054,404,615,208,893,120,717,387,306,750,936,691,325,492,548,201,070,135,531,833,963,273,177,468,177,081,270,263,577,928,899,200,342,167,009,290,552,021,804,934,495,447,718,207,787,918,112,121,662,673,676,873,010,262,038,668,557,512,992,294,351,498,845,603,399,695,802,065,546,864,461,829,119,180,374,345,920,746,172,014,024,335,189,908,611,259,445,116,953,750,027,878,726,752,943,483,220,745,953,866,524,603,651,543,097,071,113,057,038,028,281,797,656,131,368,385,212,327,780,485,867,358,895,820,964,800,788,716,083,439,787,866,141,953,372,126,587,335,240,298,424,898,966,127,967,012,479,219,892,560,422,891,781,279,419,888,466,828,000,856,827,916,291,878,113,431,094,259,846,107,018,766,633,016,580,421,919,947,617,484,016,558,162,020,958,820,771,449,262,644,942,387,677,707,819,307,421,829,321,711,722,454,587,269,702,348,314,716,462,659,011,943,011,510,869,088,797,391,198,935,894,611,079,128,068,020,798,723,914,813,727,369,675,220,010,794,965,612,345,201,072,304,403,791,536,031,331,772,280,570,856,372,178,204,350,229,193,585,617,424,310,193,833,131,174,376,421,750,066,346,416,222,787,506,802,284,495,579,054,621,609,152,530,486,148,207,870,993,810,110,268,940,440,294,074,857,757,066,644,082,273,554,832,741,196,351,312,063,887,745,953,052,089,417,037,966,998,906,118,108,112,536,828,638,951,177,501,073,738,109,561,426,148,168,809,148,623,433,905,483,155,765,995,626,293,732,678,543,317,755,749,338,738,666,937,884,021,561,748,199,087,443,038,315,466,568,533,646,251,136,796,060,307,847,008,219,007,277,249,400,614,445,285,672,466,552,661,421,279,925,108,224,741,400,590,376,136,317,829,879,862,535,597,558,640,397,748,705,262,000,177,405,089,122,135,468,554,963,925,484,755,232,253,447,387,477,724,548,793,178,444,550,403,970,302,341,414,792,098,648,109,532,795,706,589,284,915,651,975,795,293,155,297,613,388,149,951,297,597,530,453,519,017,742,001,482,967,241,457,086,922,273,259,952,329,881,744,409,425,356,167,226,734,948,023,179,315,223,770,157,167,241,765,823,228,908,544,597,374,764,225,939,059,579,467,646,720,137,878,382,071,207,197,875,371,978,649,796,296,226,028,326,681,612,709,459,831,666,172,810,646,259,314,016,943,786,710,059,862,098,810,250,586,671,802,345,052,537,281,440,933,668,683,078,820,092,644,400,262,620,883,362,833,668,089,585,892,616,132,718,855,506,390,007,507,686,051,169,803,747,132,656,286,569,867,904,065,759,562,544,187,416,285,151,334,553,586,627,410,839,017,637,596,349,918,729,058,634,996,464,885,826,071,577,498,053,939,347,036,956,907,695,254,256,751,014,158,171,884,394,704,551,251,810,483,348,520,250,392,046,743,494,507,295,152,254,249,773,739,751,987,225,647,637,218,453,857,849,561,543,180,393,923,118,112,864,415,127,436,534,212,071,930,829,606,808,834,584,118,783,606,425,950,320,668,735,509,513,183,252,304,427,046,316,410,185,647,333,861,489,468,477,308,208,652,475,243,415,638,084,125,182,380,455,455,628,918,817,222,828,032,322,715,976,157,022,968,561,123,377,237,830,761,451,604,505,946,213,074,366,965,722,092,597,846,449,438,784,554,117,877,809,389,447,902,428,209,517,871,657,618,800,337,560,441,319,034,624,865,358,732,965,536,469,883,008,300,363,233,960,423,801,201,946,665,877,502,194,420,766,018,021,918,617,612,804,326,159,254,072,949,937,587,712,953,938,405,930,960,927,024,554,416,279,743,300,422,849,127,582,934,366,958,100,840,093,073,771,142,975,668,091,446,953,142,241,809,078,164,329,041,596,855,355,148,528,836,996,637,821,378,041,529,113,123,857,756,098,948,351,194,461,035,016,700,015,186,588,476,024,686,558,098,235,952,240,933,294,170,312,454,182,693,358,454,508,585,956,163,838,649,706,434,470,539,073,551,316,781,160,260,396,063,703,454,621,637,247,067,283,398,289,180,891,445,323,097,531,847,950,835,363,437,623,712,848,281,485,880,880,507,205,908,241,529,993,841,638,904,848,001,455,989,163,554,974,024,122,260,650,074,311,772,807,210,519,700,970,080,549,729,747,369,402,899,035,971,881,651,497,876,369,745,618,653,800,116,611,801,704,931,896,876,369,277,509,069,518,333,998,509,287,767,925,642,598,266,136,138,396,755,311,142,633,589,782,990,460,723,115,843,795,544,152,133,816,364,863,436,567,985,352,444,166,762,702,043,312,292,518,421,217,325,656,256,493,301,539,207,190,977,627,414,575,173,602,160,303,396,137,961,107,270,939,169,477,864,656,961,197,406,763,308,354,341,697,660,350,733,829,141,953,134,562,624,999,296,102,016,416,396,062,116,383,683,332,156,967,960,976,720,102,250,758,432,341,888,061,625,802,577,488,143,936,602,499,581,594,896,764,256,083,128,764,286,302,061,394,251,105,382,372,289,025,409,126,576,617,628,515,787,362,956,085,941,141,507,707,231,484,306,285,435,598,594,026,463,932,850,106,446,085,594,972,801,855,647,380,827,566,427,675,665,226,717,844,823,908,778,249,522,382,239,525,855,608,061,436,386,267,959,281,178,426,641,015,399,707,480,969,961,985,362,079,826,931,437,002,191,511,327,928,623,900,957,095,249,138,083,785,149,272,678,212,752,077,701,109,144,440,215,153,567,376,912,606,388,530,280,087,831,756,873,641,135,400,851,244,326,364,487,509,359,179,194,102,850,484,441,256,239,235,844,247,549,457,592,121,353,007,978,482,198,213,903,638,162,952,213,857,164,542,621,887,207,134,376,739,558,587,446,064,435,513,012,044,668,589,533,168,951,830,803,211,090,603,249,850,903,441,843,072,014,084,083,056,592,436,669,738,158,742,038,351,804,723,042,724,972,945,970,591,061,051,581,326,583,356,803,803,563,377,619,339,875,726,216,193,091,214,741,000,312,171,646,031,101,537,330,834,664,808,067,981,751,391,299,066,178,666,923,838,363,674,604,633,136,129,889,064,504,437,483,198,753,179,200,147,223,888,880,532,310,424,254,143,654,054,376,532,440,265,707,151,964,865,238,505,958,842,820,403,960,541,461,456,110,144,142,763,044,466,193,922,011,262,651,976,178,457,496,234,141,598,550,593,162,857,334,061,987,275,435,591,523,343,128,825,325,795,810,878,545,618,782,176,784,570,063,433,799,760,530,751,612,855,035,694,497,070,136,018,275,675,903,661,125,737,446,557,249,054,010,056,827,915,678,355,185,650,063,554,826,495,978,552,180,590,603,875,138,777,407,011,867,887,845,131,421,950,456,485,270,688,983,477,088,484,058,466,656,351,957,462,521,369,909,511,351,582,412,269,739,612,018,542,080,844,145,858,684,557,607,518,671,848,669,802,622,057,946,041,851,764,808,203,199,799,262,822,928,449,296,520,202,400,722,661,559,124,371,071,386,041,618,021,718,553,591,050,606,132,370,034,784,627,555,799,351,487,486,363,836,814,514,325,202,863,873,244,446,664,815,787,426,688,836,460,430,503,652,914,620,663,350,950,954,479,934,868,559,655,058,203,359,952,860,080,238,366,494,569,720,936,320,219,013,096,330,337,095,442,401,026,770,178,563,082,376,321,583,133,978,067,990,642,048,504,345,335,846,675,001,643,564,210,566,413,518,228,309,565,072,418,166,494,485,663,696,072,648,335,664,432,164,154,396,976,015,436,171,447,703,372,905,380,092,561,244,964,599,609,375

Now, assuming we're talking about the sum of the digits (as opposed to the AMOUNT of digits), we add all of those numbers together, and we can omit all zeroes. This is how we find A.

A = 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Therefore A = 29,483

That's MUCH less intimidating. We follow the same strain of logic to compute B:

B = 2+9+4+8+3 = 26

And finally, C:

C = 2+6 = 8

Our goal doesn't involve the giant blocks of numbers; it involves A plus B plus C.

A+B+C = 29483+26+8 = 29517

If I go by your method without using a GRAAAAAAND calculator, it's going to take my whole life to do this problem, isn't there a simple way!!

: P \huge{:P}

jaiveer shekhawat - 6 years, 6 months ago

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I don't know, but now you have me wanting to look for it! Guess I'll have to figure out 2015's relationship to 29517. Edit: Exploring logarithms only offers how many digits there are in the number as opposed to what those digits are.

April Husband - 6 years, 6 months ago

their is no tecknic to slove simple way

Hareesh Batchu - 6 years, 6 months ago

How did you get A? Super computer !!

Niranjan Khanderia - 6 years, 6 months ago

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Not really. My PC solves it well under a second (about 0.0026 sec).

Arulx Z - 5 years, 9 months ago

There is no logic used to arrive at A. There is no proof to say that the value of A = 29483 A=29483 . The author expects the other solvers to take his/her word as proof.

l o g 10 ( 201 5 2015 ) = 2015 l o g 10 ( 2015 ) = 2015 × 3.30427 6658.114226 log_{10}\left(2015^{2015}\right)=2015log_{10}(2015)=2015 \times 3.30427 \approx 6658.114226

Hence, 201 5 2015 2015^{2015} has 6659 digits and it starts with the digits 1300848....(from taking 1 0 0.114226 10^{0.114226} ) But, it is not possible to get any better conclusion. Th value of A could be A = 29499 A=29499 giving B = 33 B=33 and C = 6 C=6 and A + B + C = 29499 + 33 + 6 = 29538 A+B+C=29499+33+6 = 29538

Janardhanan Sivaramakrishnan - 6 years, 4 months ago

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