It All Adds Up
We've got your numbers, but it's up to you to figure out where to put them
Row, Row, Row Your Square
1. [Not so easy] A magic square is a square grid of positive whole numbers in which every row, every column, and both main diagonals add up to the same sum. Can you arrange the numbers 1 through 9 in a 3x3 grid to make a magic square? The middle number, 5, has already been filled in.
2. [Not so easy] Can you put a 1, 2, or 3 in each square of a 3x3 grid to make a magic square? Each number will appear in three squares.
3. [Not so easy] Try finding a 3x3 magic square in which the sum of the first number minus the second number plus the third number in every row, column, and main diagonal is the same.
4. [Difficult] Can you arrange nine different positive whole numbers in a 3x3 grid so that every row, every column, and both main diagonals multiply to give the same product? The product should be as small as possible. Hint: Use the result from problem 2.
Can You Digit?
1. [Easy] Using each of the digits 0 through 9 just once, find a two-digit number and a four-digit number that add up to another four-digit number as shown at right. There are several solutions; find the smallest one possible. To get you started, we've filled in the four digits in the thousands and hundreds places, and here's the reasoning: Because no digit may be used twice, there must be a carryover from the tens to the hundreds place and from the hundreds to the thousands place. In the smallest possible solution, a 1 and a 2 must be in the thousands place. To get the carry-over into the thousands place, the two digits in the hundreds place must be 9 and 0. Can you complete the rest of the equation using each of the six remaining digits only once?
2. [Not so easy] Now find the solution with the largest possible total. (Bet you saw this coming!)
3. [Not so easy] Using each of the digits 0 through 9 just once, what are both the smallest and largest solutions for two three-digit numbers that add up to a four-digit number?
4. [Easy] Again, using each of the 10 digits just once, find a solution for eight single-digit numbers that add up to a two-digit number.
5. [Difficult] What correct equations can you assemble with the digits 0 through 9, a multiplication sign, and an equal sign? Can you find the solutions with the smallest and largest products? Hint: Neither of the smaller products can end in 0 or 1.
1. [Easy] 100 can be factored into two numbers, 4 and 25, neither of which contains a zero. Can you factor 1,000 into two numbers, neither of which contains a zero? How about 10,000?
2. [Not so easy] What is the smallest power of 10 that cannot be factored into exactly two numbers, neither of which contains a zero? Hint: You'll need a calculator for this one.
3. [Very difficult] What is the largest power of 10 that can be factored into exactly two numbers, neither of which contains a zero? Hint: This time, you should try a computer.
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