Hints and Background - but NO CHEATING!!

The low down on Magic Squares:

This is one of Benjamin Franklin's magic squares. Its rows and columns add up to 260. Every half row and half column add up to 130. The four corners and the four squares in the middle add up to 260. How many other groups of squares with sums of 260 or 130 can you find? What other patterns do you see?

Print this image or copy the numbers on a piece of paper and draw lines that connect 1 to 2 to 3 and so on. The design you'll get is wonderful! Try the same technique on the magic squares you've made with the Magic Square puzzle pieces.

There are a few tricks that make creating magic squares really easy. Click here to find out more--but only after you've tried some on your own. NO CHEATING!!

Stick Puzzle answers:

Gray dotted lines show you

which sticks to move.

Gray solid lines show you

where to move the sticks.

Welcome to the Wonderful World of Nim!

First a lesson in binary numbers. The only digits that are used in binary numbers are 1 and 0. The place values are, starting at the right and going left, 1 2 4 8 16 32 64 128 and so on. For Nim, you don't need to know how to count very far. Here's how it works:

Base Ten Binary

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15 1
10
11
100
101
110
111
1000
1001
1010
1011
1100
1101
1110
1111 (One in the one's place)
(One in the two's place, zero in the one's place)
(One in the two's place, one in the one's place: 2+1=3)
(One in the four's place, zero in the two's and one's)
(One in the four's place, zero in the two's place, and one in the one's place: 4 + 1 = 5)
(One in the four's place, one in the two's place, and zero in the one's place: 4 + 2 + 0 = 6)
(One in the four's place, one in the two's place, and one in the one's place: 4 + 2 + 1 = 7)
(8 + 0 + 0 + 0 = 8)
(8 + 0 + 0 + 1 = 9)
(8 + 0 +2 + 0 = 10)
(8 + 0 + 2 + 1 =11)
(8 + 4 + 0 + 0 =12)
(8 + 4 + 0 + 1 = 13)
(8 + 4 + 2 + 0 = 14)
(8 + 4 + 2 + 1 =15)

At the end of each player's turn, the arrangement of sticks is either "safe" or "unsafe." When your turn ends in a safe position, you can win. If your turn ends in an unsafe position, you will lose (if you're playing against someone who is an expert at Nim).

Safe positions are made unsafe by any move. Unsafe positions can become either safe or unsafe depending on the move. The first step in playing the game is to decide if the opening position is safe or unsafe. If it is safe, there is nothing you can do to make it safe so let your opponent go first. If it is unsafe, you'll be able to make it safe with your move so you should go first.

Let's look at a sample game:

Players decide to make 3 piles of sticks as shown.

In binary notation the piles have 100, 111, and 11 sticks, respectively.

Now line up the binary notations in one column:

100
111
11

Now count the number of 1s in each column. There are two in the first column, two in the second column, and two in the third column. This means that the position is safe and that means any move will make it unsafe. Let your opponent go first.

Remember that to win using the binary notation you have to take sticks so that, at the end of your turn, the number of ones in each column is even or zero.

Your opponent takes 3 sticks from the first pile.

You quickly figure the binary notations for each pile and check the number of ones in each column.

1
111
11

Now there is one 1 in the first column, two in the second column, and three in the third column. Remember, when your turn is over, there need to be an even number of ones in each column. What do you do?

You'll need to get rid of the one in the first column and the one in the third column. But if you take away all 7 sticks in the middle pile, you'll be left with one 1 in the second column. How about taking 5 sticks and leaving 2?

Now the binary notations are:

1
10
11

Since there are two ones in every column, you're safe. Your opponent takes one of the sticks from the third pile.

1
10
10

To make all the columns have either an even number or zero ones, take away the stick in the first pile.

10
10

Can you figure out how to win now? Bet you can!!

Base Ten	Binary
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15	1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 1111	(One in the one's place) (One in the two's place, zero in the one's place) (One in the two's place, one in the one's place: 2+1=3) (One in the four's place, zero in the two's and one's) (One in the four's place, zero in the two's place, and one in the one's place: 4 + 1 = 5) (One in the four's place, one in the two's place, and zero in the one's place: 4 + 2 + 0 = 6) (One in the four's place, one in the two's place, and one in the one's place: 4 + 2 + 1 = 7) (8 + 0 + 0 + 0 = 8) (8 + 0 + 0 + 1 = 9) (8 + 0 +2 + 0 = 10) (8 + 0 + 2 + 1 =11) (8 + 4 + 0 + 0 =12) (8 + 4 + 0 + 1 = 13) (8 + 4 + 2 + 0 = 14) (8 + 4 + 2 + 1 =15)