Mathina and her friend Leo hurried after the kangaroo on Logi-city’s streets. They were quite exhausted on that hot and dry summer day as they tried to keep pace with their guide. The kangaroo promised ice cream if they joined him, but they had to see all the sights of Logi-city first! Mathina and Leo hadheard of Logi-city’s famous ice cream, and could hardly wait to try it.
– There it is! – pointed the kangaroo at the huge, round ice cream stand squeezed between two skyscrapers. Leo was about to cross the street, but the kangaroo reached out after him.
– “Slow down, young man. You can’t just cross this zebra. Do you see those traffic lights on the opposite side?” – they saw one column rising, another retreating to the ground.
There was always a different number of columns rising up, and there were three lights on each of them: violet, orange, and pink, but in a different order on each column.
– “We can only cross if the three colors appear in all possible orders.” – The kangaroo smiled, noticing Leo’s amazement.
Another column rose, and the kangaroo asked, – “Do you think we can go now?”
Mathina and Leo see 3 traffic lights in these colors:
violet – orange – pink
orange – pink – violet
pink – violet – orange
Question: Do you think Mathina and Leo can cross now? Are all possible color combinations visible?
– I think we’ve got it all! – Leo shouted, and was again ready to cross the street. Now it was Mathina who grabbed his backpack and kept him from running across the cars.
– No way! – Mathina said.
– But there are three colors and three sequences. That’s good, isn’t it? After all, each of the colors comes once as first, as second and as last – Leo argued convincingly.
– And what if, for example, violet came first and then pink, instead of orange? – Mathina grinned like a little monkey.
– Leave him, Leo is probably not as good in distinguishing pink and violet as you are – said the kangaroo.
– Oh, kangaroo, don’t be like that! – Mathina got angry. – I don’t understand colors better just because I’m a girl! But I can think pretty well – and she stuck out her tongue at the kangaroo.
– Okay… The point is that Mathina is right, these three are not enough – said the kangaroo. – But then, how many lights would be enough?
Question: How many lights do you think would be enough to show all the sequences? Try to find them all by using this app.
App 1: We see a traffic light without colors that can be colored and additional lamps can be added: http://www.rimosco.it/ew/LC_S1_TrafficLight/index.htm
– I know, I know, I know! Eight is the solution, eight is the solution, kangaroo, eeeeeiiiiiight! – Leo cried and jumped up in the air. Mathina raised her eyebrows, suspicious of Leo having messed up something.
Mathina looked at Leo’s figures and said – Wrong! You’ve drawn some of them twice!
IMAGE: Traffic lights with the following combinations:
VOP
PVO
OVP
VOP
VPO
PVO
VPO
POV
Question: Leo found these color combinations below. What do you think he included twice? Put here all the sequences Leo has picked (all of them only once)!
App 2: We see the 8 traffic lights, you can rearrange them. Once you’ve put identical ones next to each other, the superfluous ones disappear http://www.rimosco.it/ew/LC_S1_TrafficLight/index.htm?v=2
VOP
PVO
OVP
VOP
VPO
PVO
VPO
POV
– You see, it’s only five now. And guess what, there’s a sixth one, too! – Mathina said contemptuously.
IMAGE: the 5 correct lights
Question: Which color order do you think is missing? (It might help to line up the lights according to the first colors, starting with violet, then pink, then orange.)
App. We see the 5 correct lamps lined up and a traffic light without colors that can be colored and additional lamps can be added. http://www.rimosco.it/ew/LC_S1_TrafficLight/index.htm?v=3
– Well, it starts with two violets and pinks, and only one in orange – Leo realized. – Do you mean we need another one starting with orange? – Wait, wait, he scratched his head. – Got it! Now, after orange comes… not violet but pink, that is: orange – pink – violet.
– Bingo! – Mathina patted Leo on his shoulder.
At this point, six traffic lights appear on the opposite side and show the sequence they created together.
IMAGE: +1 traffic lamp colored like this: OPV
The kangaroo was pleased. Mathina and Leo rushed across the street right into the huge, round ice cream store squeezed between two highrises.
IMAGE: the icecream icon on the icecream store blinks