Answers to last issue's Brain Teasers

1. Tangrams are an old Chinese game. They are a variation of a jigsaw puzzle. There are 7 pieces that form a square. To make one, just cut out a square piece of cardboard and mark off the diagonals. Also mark the centres of two sides (marked as e and i in the figure). Find the mid-point of each portion of the diagonal (marked c, f, g in the figure). Join the appropriate points and cut out the 7 pieces as shown. There are two big triangles, one medium triangle, two small triangles, one square, and just one odd-shaped parallelogram.

Now, what is the brain teaser? Re-arrange the pieces to form a rectangle.

Answer: The figure shows how the pieces can be arranged to form a rectangle. Tangrams can be made to almost any shape of your choice. Try different shapes and write to us about them.

2. I just found a number with an interesting property:

 When I divide it by 2, the remainder is 1.
 When I divide it by 3, the remainder is 2.
 When I divide it by 4, the remainder is 3.
 When I divide it by 5, the remainder is 4.
 When I divide it by 6, the remainder is 5.
 When I divide it by 7, the remainder is 6.
 When I divide it by 8, the remainder is 7.
 When I divide it by 9, the remainder is 8.
 When I divide it by 10, the remainder is 9.

It's not a small number, but it's not really big, either. Can you find it? (From Brainden.com)

Hint: If the number N gives 1 remainder when divided by 2, what is the remainder for the number N+1?
Answer: Let X be the number. From the above properties of X, one can see that, if you add 1 more to X or take away 1 from it, it will be divisible by 2. But X leaves a remainder 2 when divided by 3. So X+1 will be exactly divisible by 3. Similarly, you can see that X+1 is an exact multiple of all the numbers from 2 to 10. The smallest number with such properties must be the LCM (least common multiple) of the numbers from 2 to 10 -- which is 2520.
The simplest way to see this is to look at the list of numbers: 2 3 4 5 6 7 8 9 10.
Anything that is divisible by 9 is surely divisible by 3. So strike off 3. Similarly anything divisible by 8 is also divisible by 2 and 4. So strike off 2 and 4. But anything that is divisible by 8 and 9 (and hence 2 and 3) is also divisible by 6. So strike off 6. Also, anything divisible by 2 and 5 is also divisible by 10. So strike off 10. That leaves the numbers 5,7,8,9. The LCM is the product of these numbers. So X+1=5.7.8.9=2520. So X is one less that is, X = 2520-1 = 2519. 
Check and see that 2519 divided by the numbers 2 to 10 leaves the remainder as given.