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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/




September 2020 Brain Puzzler Solution

abcd image002

a has to be either 1 or 2 as multiplying by 4 keeps the number 4 digits.
(if a= 3 then quadrupling that number would be >12000 (5 digits)  )

if a was 1 then d would have to be 4 so our number abcd would be:
1bc4 and its reverse would be 4cb1 which cannot be as an integer multiplied by 4 has to be even.

Therefore a has to be 2.
Making d = 8.

When multiplying 2bc4 x 4  to be 8cb2, b and c cannot be too large as multiplying by 4 would then make the thousands place >8.

For example if b = 3 then even the lowest # in the 2300s,  2300 x 4 = 9200  would be too large as a = 9.

So b must be 0, 1 or 2
b cannot be 0 as 20cd x 4 = dc02 and multiples of 4 always with the last two digits as a multiple of 4.
b cannot = 2 since a is already 2 so b must = 1!

c cannot be 1, 2, or 8 as they have already been used

So abcd is now 21c8 x 4 = 8c12
If c = 0, 2108 x 4 does not end in 12
If c = 3, 2138 x 4 does not end in 12
If c = 4, 2148 x 4 does not end in 12
If c = 5, 2158 x 4 does not end in 12
If c = 6, 2168 x 4 does not end in 12
If c = 7, 2178 x 4 DOES end in 12
If c = 9, 2198 x 4 does not end in 12
c cannot be 1, 2, or 8 as they have already been used

2178 x 4 = 8712

 

 

August 2020 Brain Puzzler Solution

Question

If the probability of a false positive of a medical test (could be COVID-19) is 3%, what percent of people who test positive would actually have it?

Scenario 1: The disease is present in 1 person in 1000

Scenario 2: the disease is present in 1 person in 5


Answer

Scenario 1: 3.2% and Scenario 2: 89.3%

We need to compare the actual positives with the sum of the true positives and false positives.  In Scenario 1, if 1 in 1000 (.001) have it then 999 out of 1000 (.999) do not.

In Scenario 2, if 1 in 5 (.2) have it then 4 out of 5 (.8) do not.

 

Desmos False PositiveScenario 1:

comparing the true positives with the total positives  (true positive plus false positive)

approximately 3.2%.

Scenario 2:

comparing the true positives with the total positives (true positive plus false positive)

approximately 89%