An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
What is the smallest number with exactly 14 divisors?
Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?
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.View the archive of all weekly problems grouped by curriculum topic