# 2002 AIME II Problems/Problem 4

## Problem

Patio blocks that are hexagons $1$ unit on a side are used to outline a garden by placing the blocks edge to edge with $n$ on each side. The diagram indicates the path of blocks around the garden when $n=5$.

If $n=202$, then the area of the garden enclosed by the path, not including the path itself, is $m\left(\sqrt3/2\right)$ square units, where $m$ is a positive integer. Find the remainder when $m$ is divided by $1000$.

## Solution 1

When $n>1$, the path of blocks has $6(n-1)$ blocks total in it. When $n=1$, there is just one lonely block. Thus, the area of the garden enclosed by the path when $n=202$ is $$(1+6+12+18+\cdots +1200)A=(1+6(1+2+3...+200))A$$,

where $A$ is the area of one block. Then, because $n(n+1)/2$ is equal to the sum of the first $n$ integers: $$(1+6(1+2+3...+200))=(1+6((200)(201)/2))A=120601A$$.

Since $A=\dfrac{3\sqrt{3}}{2}$, the area of the garden is $$120601\cdot \dfrac{3\sqrt{3}}{2}=\dfrac{361803\sqrt{3}}{2}$$. $m=361803$, $\dfrac{m}{1000}=361$ Remainder $\boxed{803}$.

## Solution 2

Note that this is just the definition for a centered hexagonal number, and the formula for $(n-1)^{th}$ term is $3n(n+1)+1$. Applying this for $200$ as we want the inner area gives $120601$. Then continue as above.

## See also

 2002 AIME II (Problems • Answer Key • Resources) Preceded byProblem 3 Followed byProblem 5 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 All AIME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. Invalid username
Login to AoPS
{"ft_lang_label":"__label__en","ft_lang_prob":0.86898947,"math_prob":0.99964976,"size":1229,"snap":"2021-21-2021-25","text_gpt3_token_len":316,"char_repetition_ratio":0.13959184,"word_repetition_ratio":0.036885247,"special_character_ratio":0.270952,"punctuation_ratio":0.11020408,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9998778,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,7,null,null,null,null,null,null,null,null,null,7,null,null,null,null,null,7,null,null,null,null,null,null,null,7,null,7,null,7,null,7,null,7,null,7,null,7,null,7,null,null,null,7,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-18T12:56:16Z\",\"WARC-Record-ID\":\"<urn:uuid:d16ba3d5-0f03-470e-9af5-278ee3cbe6ba>\",\"Content-Length\":\"42280\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:19d140d6-6bb1-416f-b56d-4e55ab15e527>\",\"WARC-Concurrent-To\":\"<urn:uuid:cd83a523-c20d-4ae9-a917-19a210140f4e>\",\"WARC-IP-Address\":\"104.26.10.229\",\"WARC-Target-URI\":\"https://artofproblemsolving.com/wiki/index.php/2002_AIME_II_Problems/Problem_4\",\"WARC-Payload-Digest\":\"sha1:ACQYACT6QCHSNUADMBR5NYPC3UFYVGV6\",\"WARC-Block-Digest\":\"sha1:6ZH2WJZCJMJVYTAV44C6JI243NGT6BKO\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623487636559.57_warc_CC-MAIN-20210618104405-20210618134405-00635.warc.gz\"}"}