This problem comes from the Scottish Mathematical Council (SMC) Senior Mathematical Challenge of 2007:
“A group of seven girls—Ally, Bev, Chi-chi, Des, Evie, Fi and Grunt—were playing a game in which the counters were beans. Whenever a girl lost a game, from her pile of beans she had to give each of the other girls as many beans as they already had. They had been playing for some time and they all had different numbers of beans. They then had a run of seven games in which each girl lost a game in turn, in the order given above. At the end of this sequence of games, amazingly, they all had the same number of beans—128. How many did each of them have at the start of this sequence of seven games?”
See the Seven Girls Puzzle
This is another nice puzzle from the Scottish Mathematical Council (SMC) Senior Mathematical Challenge of 2008.
“The triangle ABC is inscribed in a circle of radius 1. Show that the length of the side AB is given by 2 sin c°, where c° is the size of the interior angle of the triangle at C.”
The diagram shows the case where C is on the same side of the chord AB as the center of the circle. There is a second case to consider where C is on the other side of the chord from the center.
See the Circle Chord Problem
This is a nice geometric problem from the Scottish Mathematical Council (SMC) Senior Mathematical Challenge of 2008.
“Mahti has cut some regular pentagons out of card and is joining them together in a ring. How many pentagons will there be when the ring is complete?
She then decides to join the pentagons with squares which have the same edge length and wants to make a ring as before. Is it possible? If so, determine how many pentagons and squares make up the ring and if not, explain why.”
See the Polygon Rings
This problem comes from the Scottish Mathematical Council (SMC) Senior Mathematical Challenge of 2008:
“S2. In Tiffany’s, a world famous jewellery store, there is a string necklace of 33 pearls. The middle one is the largest and most valuable. The pearls are arranged so that starting from one end, each pearl is worth $100 more than the preceding one, up to [and including] the middle one; and starting from the other end, each pearl is worth $150 more than the preceding one, up to [and including] the middle one. If the total value of the necklace is $65,000 what is the value of the largest pearl?”
I included the words in brackets to erase any ambiguity.
See the Pearl Necklace Problem