An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
A circle is inscribed in an equilateral triangle. Smaller circles
touch it and the sides of the triangle, the process continuing
indefinitely. What is the sum of the areas of all the circles?
Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP
have equal areas. Prove X and Y divide the sides of PQRS in the
The totals of the top row and completed main diagonal are 30 and
39 respectively and therefore the 10 consecutive numbers in
question must be 30 to 39 inclusive.
The number in the bottom right hand corner must be 1, 2, 8, 15
or 16. Taking 1 and 2 the completed diagonal would add up to less
than 30 whilst 15 and 16 produces a number greater than 39. Hence 8
must go in the bottom right hand corner. It now follows that the
last digit in the second row must be 15 since if 15 is placed in
any one of the other three vacant squares, we get a total of 45 in
the second column, 34 in the third column (same as the diagonal)
and 47 in the third row (too big). We are now left with 1, 2 and
16. End third row must be 1 to make the column less than 39 (38)
and the row total 33. This leaves 2 and 16 making the forth row
total. 16 cannot go in column 2 as this makes a total greater than
This problem is taken from the UKMT Mathematical Challenges.