This puzzle is taken from Math-E-Magic by Raymond Blum, Adam Hart-Davis, Bob Longe and Derrick Niedermann. I've owned my copy of this book for a number of years and have referered to it in the past top help with problems I've been working on separately. More recently, during a few idle moments (when there's not been enough time, energy or enthusiasm to do anything bigger) I've started solving some of the the puzzles it poses. I've even (horror of horrors) started writing IN the book (but only in pencil).
Here's the first puzzle I looked at:
Nine Coins (page 29)
Wendy got into trouble in her math class. She was sorting out money she planned to spend after school, and accidentally dropped nine coins onto the floor. The teacher was so upset that she told Wendy to stay at school until she could arrange the nine coins into at least six lines with three coins in each line. Can you do it? Wendy did, and in fact she arranged her nine coins into ten lines, with three in each line. How?
The first question - can you get nine coins into six straight lines, is fairly straightforward, especially if you realise that nine is a square number. If you arrange the nine coins into a 3x3 rectangle you can achieve six lines (three horizontal and three vertical).
If you're more careful, you can arrange them into a symmetrical rectangle, or even a square, so that the diagonals form two extra lines, bringing the total to eight. It's not ten, but it's getting us closer.
Working on the principle of increasing diagonal lines, in order to reach ten straight lines, we need to adapt the central row (or column) of coins so that they can provide the extra lines we need. By moving the two coins at the end of the central row, we can achieve the addition more diagonal lines - see below...
The new diagonal lines are shown in the paler grey colour. We have lost the vertical lines at the edges of the square, but we have gained four diagonal lines, bring our total up from eight to ten. The diagram below shows the solution with all ten lines shown: three horizontal, one vertical, two long diagonals and four short diagonals.
Here's the first puzzle I looked at:
Nine Coins (page 29)
Wendy got into trouble in her math class. She was sorting out money she planned to spend after school, and accidentally dropped nine coins onto the floor. The teacher was so upset that she told Wendy to stay at school until she could arrange the nine coins into at least six lines with three coins in each line. Can you do it? Wendy did, and in fact she arranged her nine coins into ten lines, with three in each line. How?
The first question - can you get nine coins into six straight lines, is fairly straightforward, especially if you realise that nine is a square number. If you arrange the nine coins into a 3x3 rectangle you can achieve six lines (three horizontal and three vertical).
Working on the principle of increasing diagonal lines, in order to reach ten straight lines, we need to adapt the central row (or column) of coins so that they can provide the extra lines we need. By moving the two coins at the end of the central row, we can achieve the addition more diagonal lines - see below...
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