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January 2022 Brain Puzzler Solution

Q: A snail is at the bottom of a 12-foot well. She climbs up 3 feet during the day, but slides back 2 feet each night. How many days will it take her to get out of the well?

A: 10 days
After 5 days, she is 5 feet from the bottom. After 6 days, she is 6 feet from the bottom. It will take her less than 12 days.
At the end of the 9th day, she will be 9 feet from the bottom so on the 10th day when she climbs up 3 feet, she will reach the top!

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

July 2017 Brain Teaser Solution

Q: A trader had gold coins but did not tell anyone how many she had.  If the coins are divided into two different sized groups, then 32 times the difference between the two numbers is equal to the difference between the squares of the two numbers.  How many gold coins did she have?

A: The merchant has 32 gold coins.
It is easy to check this… Let’s divide the 32 coins into two unequal numbers, say, 27 and 5. Then, 32 (27 – 5) = (27 x 27) – (5 x 5).
We can also check this by dividing the 32 coins into 30 and 2.
Then, 32(30-2) = (30 x 30) – (2 x 2).

This will work for any two numbers that add to 32.  If we call the two numbers x and y:
32 (x – y) = x^2 – y^2
So x + y = 32 and therefore y = 32 – x
Then we can rewrite the above as:
32(x – (32 – x)) = x^2 – (32 – x)^2
32(x – 32 + x) = x^2 – (1024 – 64 x + x^2)
32(2x – 32) = x^2 – 1024 + 64x – x^2  (the x^2s cancel)
64x – 1024 = 64x – 1024  🙂