All posts by mathconfidence

May 2019 Brain Puzzler Solution

Q: Let n = 2^2008 + 2008^2. What is the units (ones) digit of 2^n + n^2?

A:   This is related to the powers of two and also the last digits of squared numbers.

We first have to find n
Since the last digit of 2^(a power) has a pattern we can figure this out!

2^2008

2 to a power has a last digit pattern
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64
2^7 = 128
2^8 = 256
2^9 = 512
2^10 = 1024
2^11 = 2048
2^21 = 2097152
2^31 =

Since it is a cycle of 4, every power divisible by 4 will end with a 6…therefore 2^2008 ends with a 6…but we actually need to know the last two digits as 2^1 and 2^11 do not share the same last digit but 2^1 and 2^21 do.  So when we think about very large exponents for powers of 2, we need to know the last two digits…
2^4 ends with 16
2^8 ends with 56
2^12 ends with 96
2^16 ends with 36
2^20 ends with 76
2^24 ends with 16
2^28 ends with 56 and so on
every multiple of 20 ends with 76
so 2^2000 would end with 76
2^2004 ends with 16
2^2008 ends with 56

2008^2 =4032064
Adding up 64 + 56 = 120 so the last two digits of n would be 20

so in evaluating 2^n + n^2
First 2^n: 2^20  or 2^120 or 2^ 220 always ends in a “6

then n^2, anything with a 0 in the ones digit squared would have a ones or units digit of zero.

Therefore, the last digit would be 6.

 

 

 

 

 

March 2019 Brain Puzzler Solution

Brain Teaser March 2019

 

on the right side there are 3 pink and 3 blue
on the left side, there are 2 pink and 3 blue plus 5 green
Therefore the 5 green are equivalent to 1 pink.

On the bottom left, we can see that 1 pink = 2 green + 1 blue
so since 1 pink is 5 green and also 2 green + 1 blue,

5 green = 2 green + 1 blue
so 3 green = 1 blue or 1 blue = 3 green

Looking at the right hand side, we have 3 blue + 3 pink = 1/2(144)
If we replace blue and pnk in terms of green,
3 (3 green) + 3 (5 green) = 72
9 green + 15 green = 72
24 green = 72
1 green = 3
1 blue = 3 green = 3(3) = 9
1 pink = 5 green = 5(3) = 15

Thanks to https://solveme.edc.org/mobiles/

 

 

 

February 2019 Brain Puzzler Solution

Q: A traffic light has a cycle through green, yellow, red, green, yellow, red, etc.
Red and green are each 30 seconds with a 3 second yellow inbetween.
What is the probability that while Amir is watching during a random 3 second interval  that the color changes?

A:
9/63 = 1/7
It is a 63 second cycle.
The light changes at t=30, t = 33 and t = 63 given three three second intervals:
[27,30] and [30, 33] and [60,63] is a total of 9 seconds out of the 63 second interval