2022 AMC 10A Problems/Problem 12
Problem
On Halloween 
children walked into the principal's office asking for candy. They
can be classified into three types: Some always lie; some always tell the truth; and
some alternately lie and tell the truth. The alternaters arbitrarily choose their first
response, either a lie or the truth, but each subsequent statement has the opposite
truth value from its predecessor. The principal asked everyone the same three
questions in this order.
"Are you a truth-teller?" The principal gave a piece of candy to each of the 
children who answered yes.
"Are you an alternater?" The principal gave a piece of candy to each of the 
children who answered yes.
"Are you a liar?" The principal gave a piece of candy to each of the 
children who
answered yes.
How many pieces of candy in all did the principal give to the children who always tell the truth?
Solution 1
Consider when the principal asks "Are you a liar?": The truth tellers truthfully say no, and the liars lie and say no. This leaves only alternaters who lie on this question to answer yes. Thus, all children that answered yes are alternaters that falsely answer Questions 1 and 3, and truthfully answer Question 2. The rest of the alternaters, however many there are, have the opposite behavior.
Consider the second question, "Are you an alternater?": The truth tellers again answer no, the liars falsely answer yes, and alternaters that truthfully answer also say yes. From the previous part, we know that alternaters truthfully answer here. Because only liars and alternaters answer yes, we can deduce that there are 
liars.
Consider the first question, "Are you a truth teller?": Truth tellers say yes, liars also say yes, and alternaters that lie on this question also say yes. From the first part, we know that alternaters lie here. From the previous part, we know that there are 
liars. Because only the number of truth tellers is unknown here, we can deduce that there are 
truth tellers.
The final question is how many pieces of candy did the principal give to truth tellers. Because truth tellers only answer yes on the first question, we know that all 
of them said yes once, resulting in 
pieces of candy.
~phuang1024
Solution 2
On the first question, truth tellers would say yes. Liars would say yes because they are not truth tellers and thus will say the opposite of no. Some alternaters may lie on this question too, meaning they say yes.
On the second question, Liars would say yes because they are not alternaters and thus will say the opposite of no. Alternaters only say yes to this question if and only if they said yes to the previous question. Thus, the difference between the amount of people that said yes first and that said yes second is the amount of truth tellers, which is 
. Because it is obvious that truth tellers only say yes on the first question, our answer is .
~sigma
Solution 3
Suppose that there are 
truth-tellers, 
liars, 
alternaters who answered yes-no-yes, and 
alternaters who answered no-yes-no.
We have the following system of equations:
![\[\begin{array}{ccccccccrcccc} T &+ &L &+ &A &+ &A' &= &31, & & & & (1) \\ [0.5ex] T &+ &L &+ &A & & &= &22, & & & & (2) \\ [0.5ex] & &L & & &+ &A' &= &15, & & & & (3) \\ [0.5ex] & & & & A & & &= &9. & & & & (4) \end{array}\]](http://latex.artofproblemsolving.com/6/8/2/682827872a897e4fe2d17b1a60f19eb98649a555.png)
Subtracting 
from 
gives 
From 
it follows that 
Finally, from 
we have 
~orenbad ~MRENTHUSIASM
See Also
| 2022 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 11 |
Followed by Problem 13 | |
| 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 | ||
| All AMC 10 Problems and Solutions | ||
| 2022 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 8 |
Followed by Problem 10 |
| 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 | |
| All AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
