1. Whether a subset of the seven numbers adds up to 100 could be found by exhaustive search, but there is a more insightful approach. Five of the numbers are divisible by 3 and the other two, 26 and 35, are equal to 2 (mod 3). Hence the only way a total of 100 (= 1, mod 3) could be achieved is with both 26 and 35. Hence a solution is 26 + 35 + 39 (and it easy to see that this is the only solution).

2. As the plane moves down, the initial point where it meets the cube becomes an equilateral triangle that expands until its side is Ö2 a (where a is the side of the cube). Then each of the vertices becomes progressively truncated until, at the half-way point, a regular hexagon, of side Ö2 a/2, is obtained. This then transforms back into an equilateral triangle (oriented at 60° relative to the earlier one), which then shrinks back to a point.

3. The obvious proof is by induction.

So if the property holds for k, it holds for k +1, and it certainly holds for k = 1.

Problem Corner No. 9

1. "Tell the perpendicular drawn from the intersection of strings stretched mutually from the roots to the summits of two bamboos, fifteen and ten cubits high, standing upon ground of unknown extent". (From translation of Lilavati, a 12th century work of Hindu mathematics).

2. Someone walked a certain distance on flat land at a speed of 4 mph, then uphill at a speed of 3 mph. Retracing her exact path, she came downhill at 6 mph and walked back on the flat stretch at 4 mph again. If the total journey was 24 miles, how long did it take? (from Lewis Carroll).

3.Two chords at right angles meet a circle in A, B, C, D as shown. Prove that (in terms of arc lengths): AB + CD = BC + DA

4. Consider the 16 4-letter strings AAAA, AAAB, ..... BBBB in which each letter is either A or B. Can you arrange them in a circle so that any adjacent pair:

(a) differ in only one position (e.g. AABB and AAAB)?

(b) are the same in only one position (e.g. ABAB and BAAA)?