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 = 7 | 6-1 = 5 | |

5+2 = 7 | 5-2 = 3 | |

4+3 = 7 | 4-3 = 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 = 7 | 6-1 = 5 | |

5+2 = 7 | 5-2 = 3 | |

4+3 = 7 | 4-3 = 1 | |

3.5+3.5 = 7 | 3.5-3.5 = 0 |

**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?

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