Both a
Maths Exercise
and a
Listening Exercise!

This is a maths exercise when requiring the answers. It is also a listening exercise as follows. The facilitator reads out each situation and its questions. The answers provided are read out each time by a student. Another student listening then has to summarise the situation presented and its answers. The facilitator may wish to read through all the situations and questions first before tackling each one.

A community of twenty scientists in various locations around the world are researching into remote detection equipment into brain electrical signals that reveal individual thoughts.
One of the disciplines of the group is that, whilst they do not have to say who they have spoken to within the group, they have to declare each month how many people they have communicated with, even if they have communicated with no one. This is for a basic sense of how much the work, which may well be personal or collaborative, and sensitive and confidential, is being shared. It does not matter whether the scientists communicated by post, email, or in meetings. The returns are collected anonymously but made available.
Two of the scientists later meet and discuss the returns. One says to the other, "Someone is lying or very forgetful. One person has declared that he or she has communicated with no one - zero - whilst another is declaring that he or she has communicated with nineteen. This is impossible."
The other scientist replied, "Yes it is, although at least we have more than the minimum number of two declaring the same number."
[1] Why is a return of zero and nineteen by two different scientists impossible? Why must at least two numbers be the same?
During this period one scientist John in London sent to another scientist Janet in Tokyo his prototype printed circuits for slotting into Janet's mind examining equipment. The printed circuits had to be secure from all other people. The problem was that whilst each had a padlock and a key, they each only had one key to their own padlock. They could not risk sending the keys in the post, even separately. Nevertheless the parcel containing the padlocked box with printed circuits inside was sent from London and Janet was able to open the box in Tokyo despite it being securely locked.
[2] How was this possible?
Four scientists, Bob, Carol, Ted and Alice, ensured that each of them collaborated at least in pairs with papers held in a cupboard in the Moscow centre. They did this by quadruple locking the cupboard and handing out more than one key to each of them in unique combinations.
[3] How many keys did each receive so at least two had to work on the papers and in what manner did any of them miss out on receiving keys?
Some scientists nevertheless used code to secure their communications. One scientist, Joe Ninety, had the habit of sending single words in code. It said:
This equipment will need to reproduce like a sleep to be effective around the world. Yours, Nine.
The other scientist Peter Rabbit knew that only ever one word was encoded by Joe's often used reverse shift letters system, and that his colleague was not called Nine but Ninety (even by nickname). Nevertheless, Peter could not work out the sentence.
[4] Which single word was encoded so that the sentence makes more sense? How was it done?
Scientists discover that a single usable bubble of memory needs the capture of thirty microbubbles in a row. Equipment produced by one scientist captures three microbubbles in one nanosecond, but, to then gain another three, two of the gained microbubbles are lost (rather like forgetting) and need to be recaptured in the next three microbubbles captured.
[5] How many nanoseconds does it take to gain just one usable bubble and what is the explanation?


See Flannery, S., Flannery, D. (2000), In Code: A Mathematical Journey, London: Profile Books, 237, 242-243, 245-246.