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November 2020 Brain puzzler solution

This puzzler has many solutions and is part of the Open Middle collection of problems. These questions stretch student thinking as they offer multiple solutions and are not PhotoMathable.

Place a digit in each box to make a true statement.
You can use 1 to 9 but only one time each.
How many solutions can you find?
(please do not use a calculator)

It seems that there are 10 solutions:
3/15 = 0.2
7/35 = 0.2
9/45 = 0.2
6/15 = 0.4
6/12 = 0.5
7/14 = 0.5
8/16 = 0.5
9/18 = 0.5
9/15 = 0.6

Here were the hints:
How does choosing the digits for the denominator affect the decimal value?
How might choosing the digit for the decimal make finding the digits for the fraction easier?

Converting a Fraction to a Decimal

October 2020 Brain puzzler solution

Q: FUN is a three digit number where F, U, N are single digits (not including 0).If FUN = F! + U! + N!, then what is the number FUN?
(where ! = factorial).

A: 145 = 1! + 4! + 5!

Because it is a three digit number, none of the digits can be more than 6 as 7! is 5040 and is therefore four digits. This means that we only need to try 1 2 3 4 5 and 6. But 6! is 720 and we only have 1 2 3 4 5 6 so now 6 is disqualified.
This leaves us with only 1 2 3 4 5.

5! is 120 (5 x 4 x 3 x 2 x 1)
4! is 24 (4 x 3 x 2 x 1)
3! is 6 (3 x 2 x 1)
2! is 2 (2 x 1) and
1! is 1

In order for the number to add up to three digits, 5! has to be one of the components. Our answer will definitely start with F=1 and therefore one of the other digits is 1.

Now we have two out of 3 digits, F = 1 and N = 5 and 1! + 5! = 121 (our subtotal)

Now we will try adding U! 2!, 3! and 4! to our subtotal.

1! + 2! + 5! = 1 + 2 + 120 = 123 not equal to FUN of 125
1! + 3! + 5! = 1 + 6 + 120 = 127 not equal to FUN of 135
1! + 4! +5! = 1 + 24 + 120 = 145 and is EQUAL to FUN of 145.

A shoutout to Colin Porteus who amended the solution for July’s Brain Puzzler — thanks for your sharp mind and eyes.
https://mathconfidence.com/2020/07/27/july-2020-brain-puzzler-solution/