A number of eggs dyed various colors were hidden for an egg hunt. How

Official Solution:

A number of eggs dyed various colors were hidden for an egg hunt. How many eggs in total were hidden?

Initially, we can't really rephrase the question. We are asked for the total number of eggs hidden for a hunt.

Statement 1: INSUFFICIENT. This statement tells us 2 facts. Using \(x\) to represent the unknown integer, we can write the following:

Red Eggs \(= x^2\)

Total Eggs \(= 24x\)

However, we have no way of determining x, so this statement is not enough.

Statement 2: INSUFFICIENT. This statement tells us the following:

Non-Red Eggs \(= 143\)

By itself, we cannot hope to know how many eggs were hidden in all.

Statements 1 and 2 together: INSUFFICIENT. We know the following:
\(\text{Red} + \text{Non-Red} = \text{Total}\)
\(x^2 + 143 = 24x\)

We can rearrange this quadratic equation, setting one side equal to 0:
\(x^2 - 24x + 143 = 0\)

At this point, we can stop if we study the equation closely. The factored form of the equation must be as follows:
\((x - ...)(x - ...) = 0\)

The reason is that the middle term (\(-24x\)) is negative, while the constant term (143) is positive. This means that the factored form on the left must have two minus signs.

As a result, we expect two positive solutions for \(x\). In fact, we could have just one positive solution, if the equation factors into something like this: \((x - ...)^2 = 0\). However, that would require the constant term (in this case, 143) to be a perfect square, since \(x\) is an integer. (For instance, if the original equation were \(x^2 - 24x + 144 = 0\), it would factor to \((x - 12)^2 = 0\), and \(x\) would have just one possible value, 12.) Thus, there are two possible values of \(x\).

Alternatively, we could simply factor \(x^2 - 24x + 143 = 0\). Since \(143 = 11 \times 13\), we have the following:
\((x - 11)(x - 13) = 0\)

\(x = 11\) or \(x = 13\).

Thus, there are two possible values for \(x\), leading to two possible total numbers of eggs. Even together, the statements are not sufficient.

Answer: E.

\({\rm{Total}}\,\,\left\{ \matrix{\\
\,{\rm{red}}\,\,:\,\,\,R \ge 1\,\,{\mathop{\rm int}} \hfill \cr \\
\,{\rm{not - red}}\,\,:\,\,\,N \ge 1\,\,{\mathop{\rm int}} \hfill \cr} \right.\)

\(? = R + N\)

Let´s go straight to (1+2): a BIFURCATION will prove the correct answer is (E).

\(\left( {1 + 2} \right)\,\,\left\{ \matrix{\\
N = 143 \hfill \cr \\
R = {M^2}\,\,,\,\,R + N = 24M\,\,\,\,\left( {M \ge 1\,\,{\mathop{\rm int}} } \right)\,\,\,\left( * \right) \hfill \cr} \right.\)

\(\left( * \right)\,\,\, \Rightarrow \,\,\,N = 24M - {M^2} \Rightarrow \,\,\,\,11 \cdot 13 = 143 = M\left( {24 - M} \right)\,\,\,\, \Rightarrow \,\,\,M = 11\,\,{\rm{or}}\,\,M = 13\)

\(\left\{ \matrix{\\
\,M = 11\,\,\,\,\, \Rightarrow \,\,\,\,\,R = {11^2}\,\,\,\,\, \Rightarrow \,\,\,\,\,? = {11^2} + 143 \hfill \cr \\
\,M = 13\,\,\,\,\, \Rightarrow \,\,\,\,\,R = {13^2}\,\,\,\,\, \Rightarrow \,\,\,\,\,? = {13^2} + 143 \hfill \cr} \right.\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.