An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
Choose any three by three square of dates on a calendar page...
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Let the numbers at two of the other vertices be $ u$ and $v$, as shown in the diagram. The three faces sharing the vertex labelled with the number 1 all have the same sum. Then $1+v+u=1+5+u$ and so $v=5$.
Similarly, $1+v+5 = 1+v+u$ so $u =5$.
Hence the sum for each faces is $1+5+5=11$, and we see that the number at the bottom vertex is $1$.
The total of all the vertices is $1+5+5+5+1=17$.
This problem is taken from the UKMT Mathematical Challenges.