# Strategy Tips

Some of the larger boards in Factor Find can look pretty overwhelming at first, and you might be tempted to reach for a calculator. But fear not, this article provides three strategy tips for efficiently solving more difficult puzzles in Factor Find!

The best way to get better at Factor Find is by playing. So, let’s try to find the number 99225 on this board: The intuitive approach is to think of this as two separate problems:

1. First, we would need to find a group of numbers between 1 and 9 that multiply together to make 99225.
2. Second, we would need to find that same combination of numbers on the board.

This approach will work for simpler puzzles, but even mathematicians would have trouble dividing 99225 into small numbers without a calculator or pen.

Instead of solving two separate (and hard!) problems, we can cleverly use the board to combine these two problems into one. By using the following three tips, we avoid having to do any long division while still finding 99225 quickly and reliably!

### Tip 1: Divisibility Rules

The Divisibility Rule for a number is a simple way to determine whether or not it can be used to make a target number. For example, we can use the Divisibility Rule for 5 to determine that 99225 contains 5, which means that we can immediately eliminate any rows, columns, and diagonals that don’t contain at least one 5.

Here are some Divisibility Rules that you might find useful while playing Factor Find:

• 1: Any number can contain 1.
• 2: If a number’s last digit is even (0, 2, 4, 6, or 8), it contains 2.
• 844 contains 2 (ends in an even number)
• 99225 has no 2 (ends in an odd number)
• 3: If the sum of a number’s digits can be divided by 3, it contains 3.
• 99225 contains 3 (9 + 9 + 2 + 2 + 5 = 27. 27 / 3 = 9)
• 844 has no 3 (9 + 3 + 4 = 16. 16 can’t be divided by 3)
• 4: If a number’s last two digits can be divided by 4 (or 2 twice), it contains 4.
• 844 contains 4 (44 / 2 = 22. 22 / 2 = 11)
• 99225 has no 4 (25 can’t be divided by 4)
• 5: If a number’s last digit is 0 or 5, it contains 5.
• 99225 contains 5 (Ends in 5)
• 844 has no 5 (Doesn’t end in 0 or 5)
• 6: If a number contains 2 and 3, it contains 6.
• 3312 contains 6 (Contains 2 and 3; see above)
• 99225 has no 6 (Doesn’t contain 2)
• 7: If twice a number’s last digit subtracted from its other digits can be divided by 7, it contains 7.
• 99225 contains 7 (9922 – 10 = 9912. 991 – 4 = 987. 98 – 14 = 84. 8 – 8 = 0)
• 844 has no 7 (84 – 2*4 = 76. 7 – 2*6 = -5)
• 8: If a number’s last 3 digits can be divided by 8 (or 2 three times), it contains 8.
• 8448 contains 8 (448 / 2 = 224. 224 / 2 = 112. 112 / 2 = 56)
• 99225 has no 8 (225 can’t be divided by 2)
• 9: If the sum of a number’s digits can be divided by 9, it contains 9.
• 99225 contains 9 (9 + 9 + 2 + 2 + 5 = 27. 27 / 9 = 3)
• 844 has no 9 (8 + 4 + 4 = 16. 16 can’t be divided by 9)
• 25: If a number’s last two digits are 00, 25, 50, or 75, it contains 25 (two 5’s).
• 99225 contains 25 (ends in 25)
• 844 has no 25 (ends in 44)

Using these rules, we see that we must make 99225 using a row, column, or diagonal that contains at least two 5’s, one 7, one 9, and no 2’s, 4’s, 6’s, or 8’s. This is enough information to start looking for a solution on the board, and we haven’t done any long division!

### Tip 2: Focus on 5 and 7

We now know to find 99225 by looking for runs of numbers that don’t contain any 2’s, 4’s, 6’s, or 8’s but do contain 3’s, 5’s, 7’s, and 9’s.

However, that task can still be difficult because a 9 may be represented as a combination of 3’s and/or 6’s in addition to being represented as 9. For example, 3 * 3, 3 * 6, and 6 * 6 all contain a 9. Similarly, if a target number contains an 8, its solution on the board might not include an 8 because 2 * 2 * 2, 6 * 4, etc. all contain 8s.

For this reason, a good approach is to look for 5’s and 7’s first. Because the set of numbers from 1 to 9 only has one number that contains 5 (5) and one number that contains 7 (7), if a target number contains a 5 or 7, then its solution on a Factor Find board will include a 5 or 7.Similarly, if a target number doesn’t contain 5 or 7, a good first step is to look for runs of numbers without a 5 or 7.

### Tip 3: Minimum Length Rule

Using the first two tips, we can narrow down the possible solutions to 99225 to three runs of numbers on the board: At this point, it would be okay to start swiping and guess all three possibilities until you get the right one. However, we can use one more strategy to avoid guessing.

99225 is a large number, so a lot of numbers have to be multiplied together to make it… more than just two 5’s, a 7, and a 9. More specifically, there must be one number in the solution for each digit in the target number.

Even more specifically, the biggest number we could make on a Factor Find board with five numbers is only 9^5 = 59049, so the solution to 99225 actually must contain at least six numbers. Additionally, we don’t count 1’s because multiplying a number by 1 doesn’t change it.

Here is the biggest number we can make on a Factor Find board with each amount of numbers (not including 1’s):

• 1: 9^1 = 9 (~10)
• 2: 9^2 = 81 (~100)
• 3: 9^3 = 729 (~1,000)
• 4: 9^4 = 6,561 (~10,000)
• 5: 9^5 = 59,049 (~100,000)
• 6: 9^6 = 531,441 (~1,000,000)
• 7: 9^7 = 4,782,969 (~10,000,000)

You don’t have to remember all of those numbers to play Factor Find. The main takeaway from this is that very big numbers are actually very easy to find because they take up almost the entire row, column or diagonal. This means that knowing a little information about a number is often enough to eliminate all but one or two rows as possibilities without doing long division.

### Conclusion

Using the Minimum Length Rule, we can see that 99225 must be made with at least six numbers (not counting any 1s).

Of the three remaining candidates, only one has at least six numbers. So, we can find 99225 without guessing or doing too much mental math! Hopefully these tips will make playing Factor Find more fun and rewarding. Enjoy!