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Combinatorics Difficulty 1.3 Junior Find the answer Canada

The sequence of the five symbols $\bigcirc$ $\blacktriangleleft$ $\boxtimes$ $\bigtriangleup$ $\bigstar$ repeats to form the pattern:

$\bigcirc$ $\blacktriangleleft$ $\boxtimes$ $\bigtriangleup$ $\bigstar$ $\bigcirc$ $\blacktriangleleft$ $\boxtimes$ $\bigtriangleup$ $\bigstar$ $\cdots$

If the pattern is continued, the 23rd symbol in the pattern is

Pick one

Solution

Solution 1:

The sequence repeats every $5$
symbols, and $23=4\times5+3$ . Thus,
the sequence of $5$ symbols repeats
$4$ times, and $3$ additional symbols follow. The 3rd
symbol in the sequence is $\boxtimes$ , and thus the 23rd symbol is
$\boxtimes$ .

Solution 2:

If the sequence of $5$ symbols is
repeated $4$ times, then $5\times4=20$ symbols are written.
Following this, $3$ more symbols are
needed to reach $23$ , and since the
3rd symbol in the sequence is $\boxtimes$ , the 23rd symbol is $\boxtimes$ .

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.