## Task 1 - 300811

Axel lets Jörg roll a red, a blue, and a yellow die. Without seeing the rolled numbers, he then says:

"Double the number on the red die, add 8 to it, and multiply the sum by 50! Remember the result! Now add 10 to the number on the blue die and multiply the sum by 10! Finally, form the sum of the product just obtained, the previously remembered result, and the number on the yellow die. If you tell me this sum, I can tell you the number rolled on each of the three dice."

a) Choose three possible numbers and perform the calculations as specified!

b) Describe how to determine the numbers from the sum given at the end of the calculations! Explain why the correct numbers can always be found according to your description!
a) If the numbers on the red, blue, and yellow dice are approximately 4, 1, and 6, respectively, the calculation leads to $(2-4+8) \cdot 50=800, (1+10) \cdot 10=110$, and thus to the sum $800+110+6=916$.

b) The chosen numbers on the red, blue, and yellow dice are:

The hundred's digit reduced by 5, the ten's digit, the unit's digit.

This can be explained as follows: If $r, b, g$ are the numbers, the calculation leads to $(2 \cdot r+8) \cdot 50 = 100 r + 400, (b+10) \cdot 10 = 10 b + 100$, and thus to the sum

$$
100 r + 400 + 10 b + 100 + g = 100 r + 10 b + g + 500
$$

If one reduces the hundred's digit in this sum by 5, i.e., subtracts 500, the number $100 r + 10 b + g$ is obtained, which is the number with the digits $r, b, g$.