Thursday, June 25, 2015

A cheating friend (or funny dice)

who: J1 and J2
when: at dinner

The basic puzzle

In class recently, we've been using a warm-up puzzle where we tell the students we have two secret numbers and give them two clues: the sum of the numbers and their difference. The challenge is to find the two secret numbers.

This is a simple algebra problem that can also be attacked easily by guess-and-check (for small values) or by a systematic table of trials. These are the methods being used by the 2nd and 3rd graders, so the puzzles result in a lot of arithmetic practice as well as reinforcing different number bonds.

Our twist

Over dinner, I told the kids that a friend gave me a similar puzzle, but with a slight twist. This friend rolled a 6-sided dice twice and told me the sum and the difference of the values. Sounds the same, but easier, right...?

Unfortunately, our friends are all a bit tricky.
First clues
So, the first clues my friend gave me were that her two numbers sum to 12 and have a difference of 4. The two talked through these clues and found the answer (which I'll leave to you).

But wait: these were supposed to come from dice throws!?

Second clues
I told them that I'd gotten the same conclusion and had challenged my friend. When confronted with this result, the friend said, "oh, there was a mistake. Actually, the difference was 12 and the sum was 4."

After a minute, but before finding a solution, the J2 realized that there was something fishy about this one, two. How can two dice rolls have a difference of 12?!!

Are there two numbers that fit the clues, even if they aren't from dice rolls? Yes, J1 and J2 found them.

Third clues
So, the friend was tricking us again. So what were the actual values? This time, the friend claimed that the sum is 7 and the difference is 2.

At this point, both were highly suspicious, but also very curious to see what would go wrong. This time, they made a table with number bonds of 7 and the differences. It looked something like this:
7+0 = 7        7-0 = 7
6+1 = 76-1 = 5
5+2 = 75-2 = 3
4+3 = 74-3 = 1

At this point, they noticed a troubling pattern in the differences: always odd. J2 suggested adding another line to the table:
7+0 = 7        7-0 = 7
6+1 = 76-1 = 5
5+2 = 75-2 = 3
4+3 = 74-3 = 1
3.5+3.5 = 73.5-3.5 = 0
Ah, now we've got an even value! From there, it took another couple of minutes to find the combination of numbers that add to 7 and have a difference of 2.

Final answer, Final puzzle
Once again, the friend seems to be tricking us: these numbers aren't on a regulation 1 to 6 die either! It was time for the final revelation: actually, all of these were from my friend's dice throws, but they just had very strangely marked dice.

Looking over our work, we saw that we had only come up with 5 different numbers as answers in our puzzles. We confirmed that, yes, the dice was 6 sided, so what was the value of the last side?

This was the last clue from the friend: the average value of the faces on her die is the same as the average value for a regular die.

So, how are the sides of my friend's die marked?

No comments:

Post a Comment