This is a test of Brilliant features

This is a solution.

Understand the problem: Understand the goals, Restate the problem

• For simplicity, we will use $\overline{AB}$ to denote the 2-digit number that is equal to $10A + B$ . It is not equal to $A \times B$ .
• The problem states that $\overline{3X} + 7 = 41$ .

Solution strategy: Choose a strategy
We want to isolate the value of $X$ . Is there a way that we can move terms around to solve for $X$ ?

Carry out the plan: Implement the strategy
If we subtract 7 from both sides, we obtain $\overline{3X} = 41 - 7 = 34$ .
If we subtract 30 from both sides, we obtain $X = 34 - 30 = 4$ .

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Understand the problem: Understand the goals, restate the problem

• For simplicity, we will use $\overline{AB}$ to denote the 2-digit number that is equal to $10A + B$ . It is not equal to $A \times B$ .
• The problem states that $C + \overline{CC} = \overline{D4}$ .
• The problem states that $C \neq D$ .

Solution strategy: Investigate the conditions, Choose a strategy
It is not immediately clear how the condition $C \neq D$ is to be used. Let's leave it aside for now.
If we look at the units column, we get that $C + C$ ends in 4. This gives us alot of information, and could tell us what $C$ is.

Carry out the plan: Integrate the information, Check for unused information

• Since $C + C$ ends in 4, and $C + C < 18$ , we either have $C + C = 4$ or $C + C = 14$ .
• This gives us $C = 2$ or $C = 7$ , and we have to check the cases.
• If $C = 2$ , then $\overline{D4} = 2 + 22 = 24$ . This gives us $D = 2$ , which contradicts the condition that $C \neq D$ .
• If $C = 7$ , then $\overline{D4} = 7 + 77 =84$ . This satisfies $C \neq D$ . Hence we have $C = 7$ .

Food for thought: Alternative solution strategy
Recall that we initially didn't know how to use the condition that $C \neq D$ . Actually, there is another approach which allows us to do so. If we consider the tens column and account for possible carry-over, then we either have $C = D$ or $C = D+1$ . But since $C \neq D$ , hence we must have $C = D + 1$ . The carry-over tells us that $C + C = 10 + 4$ , and thus $C = 7$ . This is an ingenious approach that utilizes the given conditions in an alternative manner.

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Understand the problem: Gather the information, Restate the problem

• We are given some information about the brothers and want to find the youngest.
• Let's display the information in a clearer manner. Construct a table with the brothers as rows and the columns as relative ages. We use a 1 to indicate that the corresponding brother is of that age, and 0 otherwise.

$\begin{array} { c | c | c | c | } & \text{Oldest} & \text{Middle} & \text{Youngest} \\ \hline \text{ Kevin} & 1 & & \\ \hline \text{Nicholas} & 0 & & \\ \hline \text{Joseph} & & & 0 \\ \hline \end{array}$

Solution strategy: Choose a strategy
Each row (and column) must have exactly one 1, and the rest are 0's. Based on that rule, let's fill in as much information as possible.

Carry out the plan: Integrate the information

• Consider Kevin's row. Since there is already a 1, the rest must be 0.

$\begin{array} { c | c | c | c | } & \text{Oldest} & \text{Middle} & \text{Youngest} \\ \hline \text{ Kevin} & 1 & 0 & 0 \\ \hline \text{Nicholas} & 0 & & \\ \hline \text{Joseph} & & & 0 \\ \hline \end{array}$

• Consider the Youngest column. Since there is only 1 blank cell, and no 1's as yet, it must be a 1.

$\begin{array} { c | c | c | c | } & \text{Oldest} & \text{Middle} & \text{Youngest} \\ \hline \text{ Kevin} & 1 & 0 & 0 \\ \hline \text{Nicholas} & 0 & 1 & \\ \hline \text{Joseph} & & & 0 \\ \hline \end{array}$

• Hence, we can conclude that Nicholas is the youngest!

Food for thought: Alternative solution strategy, Follow up question

• Which rows/columns should we focus on filling out? How would you categorize them?
• Can you find another sequence of steps to arrive at the conclusion?
• Can you determine who the middle child is?

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Understand the problem: Gather the information, Restate the problem

• We know that these men are called Andrew and Bob, and have to figure out who Andrew is.
• We are given that at least one of them lied. It's possible that only one lied, or that both lied.

Solution strategy: Investigate the conditions
What can we conclude if we know that the person lied?
If the man in the $\color{#3D99F6} {\text{blue} }$ shirt lied, then he is not Andrew, hence he is Bob.
If the man in the $\color{#D61F06} {\text{red}}$ shirt lied, then he is not Bob, hence he is Andrew.

Solution strategy: Choose a strategy

We now consider cases based on who told the truth, and see what conclusions we can draw.

Since we only have 1 possible case, hence Andrew is the man in the red shirt.