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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

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  🙂

 

 

Get the Math and the Points! Jan 2016 CC Alg I #20

Jan 2016 Alg I 20
This is a transformation both horizontal and vertical translation or moving!
The – 2 moves down the entire graph by 2 as it is faithful and does what we think it will do
Only answers (1) and (3) have been moved down
But the (x + 1) part is the horizontal shift and does the opposite of what it looks like so it will shift left
The easiest way to do this is to pick a point on the original graph like (2,3)
Move it one unit left and two units down to (1,1)
Which of the two graphs for answers (1) and (3) go through (1,1)?

 

 

 

 

 

March 2016 Brain Teaser Solution

Q:  What whole number can be added to 36, 300 and 596 so they all become perfect squares?

A: 925

36 + 925 = 961 (31^2)
300 + 925 = 12225 (35^2)
596 + 925 = 1521 (39^2)

Can try with a list of perfect squares subtract 36 and see if it is a perfect square.  If it is, now subtract 300 and check that answer.  If that one works, then try subtracting 596.
The lowest perfect square we can start with is 625 (25^2) as it is the first one larger than 596.
625 – 596 = 29 which is not a perfect square so we move to the next perfect square, 676 (26^2).

 

 

This was a modified problem from a 1989 contest:
https://www.artofproblemsolving.com/wiki/index.php?title=1989_AIME_Problems/Problem_7