Three students had collected some data on the wingspan of some bats. Unfortunately, each student had lost one measurement. Can you find the missing information?
Can you number the vertices, edges and faces of a tetrahedron so
that the number on each edge is the mean of the numbers on the
adjacent vertices and the mean of the numbers on the adjacent
Ann thought of 5 numbers and told Bob all the sums that could be made by adding the numbers in pairs. The list of sums is 6, 7, 8, 8, 9, 9, 10,10, 11, 12. Help Bob to find out which numbers Ann was thinking of.
Here is another excellent solution from Andrei
Lazanu, age 12, School no. 205, Bucharest, ROMANIA. Well done
I started with a1 = 9 and b1 = 3, and I used the recursive
relations from the problem. I obtained the following values (to
three decimal places):
I represented the successive "a" terms and "b" terms on a number
line and I saw that both of them go toward the same value. Using 3
decimals, practically, after the 3-rd iteration (a4 and b4) the
same value is reached.
[Note that each time the geometric mean is less than the
arithmetic mean. The arithmetic means are decreasing (can you
explain why?) and the geometric means are increasing (why?) so they
Now, I choose my numbers, and I looked for more distant numbers
to see how fast they arrive at the same value: a1 = 2 and b1 =
I wrote the following programme in Matlab, to generate the terms
of the sequences "a" and "b":
% arithmetic and geometric means
I obtained the following values: for the "a" (left column) and
"b" (right column) sequences respectively (all values must be
multiplied by 104).
I saw that after the 5-th iteration, to the precision used, the
two sequences arrived at the same value.
[You may like to use a spreadsheet to do this