# Periodic Function - Repeating Pattern Problem

For the following question:

A necklace is made by stringing N individual beads together in the repeating pattern of Red Bead , Green Bead , White , Blue and Yellow Bead. If necklace begins with a RED Bead and ends with a white bead , then N could be A)$16$ B)$32$ C)$41$ D)$54$ E)$68$

I think the answer should be 41 but my text says its 68. Is this a misprint?

Here is how I am solving it:

Let n= $1$ for Red and let n=$2$ for green and so on. Now for the answers if n=$16,32,54,68$ they are divisible by $2$ so they would end up in green but for n=$41$ we could write $40+1$ so its Green and then White.

Also one more question , say if we had n=40 that is divisible by both n=2(Green) and n=4(Blue) and also by n=5 (Yellow) then the 40th one which would it be Blue , Green or Yellow ?

-

Ok so we need to find $N$. First off, there are 5 beads in the pattern. Red is bead number 1, and yellow is bead number 3. Since we are ending with bead number 3, and there are 5 beads in the pattern, then 5x=N - 3, where x must be an integer. What does this equation mean? You must subtract 3 from N, and the result must be divisible by 5. 41-3 = 38, which is not divisible by 5, 68-3 = 65 is divisible by 5.

Another way to look at this is see that bead 1 is red. Continuing down the cycle, bead 6 is also red. Every 5th bead onwards is read. Beads 11,16,21..36,41 are red. So bead 41 is also red, which would mean that there can't be 41 beads in the pattern. Bead 66 is also red. So bead 67 would be green, and 68 would be white.

As for your second question, you have to understand that it does not work like that. Suppose you have a pattern ABCDE of repeating items. You know that the pattern starts at A, and you have your N value. Let i = the index of any letter in that sequence. For A, i = 1; for B, i = 2; and so on. If you want to test if the letter at the i-th index corresponds to a certain N value, then you need to see if (N-i)/(i) is an integer number. For example, if you want to see if the 43rd bead is blue, where i = 4, then do (43-4)/4 = 39/4, which is not an integer answer, so the 43rd bead is not blue.

-
Is their any way by which we could tell whether the $40$th bead would be Green,Yellow or Blue ? Is my method flawed ? – MistyD Aug 26 '12 at 16:41
@MistyD I just edited my answer, please look at the last part. – mathguy Aug 26 '12 at 16:42
@MistyD: The pattern you are given says every fifth bead is yellow. You are right that you could form patterns of two or four that would end on the 40th bead, but that is not what the problem says here. – Ross Millikan Aug 26 '12 at 16:42
@MistyD You mean assuming that the beads start at red and the pattern is still red,green,white,blue,yellow? – mathguy Aug 26 '12 at 16:51
@MistyD if so, take the modulus of the number 40. Since there are 5 numbers in the cycle, you would do 40 mod 5, which gives you an answer of 0. If the answer is 0, then its red, 1:green, 2:white: 3:blue, 4:yellow – mathguy Aug 26 '12 at 16:53

Arrange the beads in rows of five:

$$\begin{array}{r} R&G&W&B&Y\\ \hline 1&2&3&4&5\\ 6&7&8&9&10\\ 11&12&13&14&15\\ 16&17&18&19&20\\ \vdots&\vdots&\vdots&\vdots&\vdots \end{array}$$

• In each column all of the numbers have something in common; what is it? HINT: What happens when you divide them by $5$?

• Which column contains the number that you want?

• Which column contains the number $40$?

-

Hint: you have a cycle of five beads, red through yellow, then start again. At the end you do a partial cycle of how many beads? So the total number of beads must be xxx more than a multiple of yyy.

-