# Challenge Riddles and Puzzles: Extend the following to a valid equation

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#### BvU

Homework Helper
139. This time you are the
start with a total of 21 and add 2 at each turn -- no way to get an even number

#### fresh_42

Mentor
2018 Award
139. This time you are the
start with a total of 21 and add 2 at each turn -- no way to get an even number
It is hard to compete with codes, smart people and quick stuff ... Should have taken a bit longer.

BvU

#### fresh_42

Mentor
2018 Award
140. Can you decode the password?

D131

#### BvU

Homework Helper
It is hard to compete with codes, smart people and quick stuff ... Should have taken a bit longer.
How come you have such a high like percentage ?

#### fresh_42

Mentor
2018 Award
141. Can you separate the bulls with two square fences?

D131

Gold Member
#141

#### fresh_42

Mentor
2018 Award
142. 1600 bananas are distributed among 100 monkeys. Individual monkeys can also go out empty-handed. Prove that there are always four monkeys with the same number of fruits!

D131

#### BvU

Homework Helper
142.
Cheapest is to give three monkeys nothing, three monkeys one banana each, three two each etc. By the time you have given the 97th monkey sixteen bananas you are out of bananas.

#### fresh_42

Mentor
2018 Award
142.
Cheapest is to give three monkeys nothing, three monkeys one banana each, three two each etc. By the time you have given the 97th monkey sixteen bananas you are out of bananas.
I don't get it. 16 bananas would be given for the 51st monkey. And 3(0+1+2+...+16)=408 leaves me with 49 monkeys and 1192 bananas. Applying your scheme, I will have 112 bananas left before I turn to monkey 97. Number 96 received 31 bananas.

#### BvU

Homework Helper
Forgot the three that get nothing - poor sods. Monkey 99 gets 32 bananas and THEN there's only 16 of them yellow thingies left.

The idea was good - the execution slightly less ....

#### fresh_42

Mentor
2018 Award
143. How many numbers are there that contain their length as a digit? Example: 1024 has the length 4 and also contains the number 4. Is there a solution without a brute force code?

D131

#### DrClaude

Mentor
Take $n$ to be the length of the number. There are $9 \times 10^{n-1}$ numbers of length $n$: 9 possibilities for the first digit, since it cannot be zero (*), and 10 for each subsequent digit. The count of numbers that do not have the digit $n$ in them is $8 \times 9^{n-1}$ (same as the previous formula, but with one less possibility at each position). Therefore, there are
$$\sum_{n=1}^{9} \left( 9 \times 10^{n-1} - 8 \times 9^{n-1} \right)$$
numbers that contain their length as a digit. Using a calculator, I get this sum as 612579511.

Edit: (*) zero can be the first digit for $n=1$, but the formula obtained works also in that case.

#### fresh_42

Mentor
2018 Award
144. I know you - and me - don't like them very much, but from time to time ...
What is the next row?
1
11
21
1211
111221
312211
13112221
1113213211
31131211131221
...

D132

#### TeethWhitener

Gold Member
#144
13211311123113112211

#### fresh_42

Mentor
2018 Award
145. What is the smallest natural number with a multiplicative persistence of $5$ in the decimal system?
The multiplicative digit root is the repeated process of multiplying all digits, e.g.
$$3784 \longrightarrow 672 \longrightarrow 84 \longrightarrow 32 \longrightarrow 6$$
Tme multiplicative persistence is the number of arrows, iterations. For $3784$ it is $4$.

D132

#### DavidSnider

Gold Member
145. What is the smallest natural number with a multiplicative persistence of $5$ in the decimal system?
The multiplicative digit root is the repeated process of multiplying all digits, e.g.
$$3784 \longrightarrow 672 \longrightarrow 84 \longrightarrow 32 \longrightarrow 6$$
Tme multiplicative persistence is the number of arrows, iterations. For $3784$ it is $4$.

D132
679

#### fresh_42

Mentor
2018 Award
146. A newly combined class has $33$ students. Each of them introduces themselves by first name and family name. The kids recognize, that some have the same first name and some even the same family name. So every kid writes on the blackboard how many others have the same first name, and how many the same family name, not counting themselves. At the end there are $66$ numbers on the board, and every number $0,1,2,\ldots, 9,10$ occurs at least once.

Are there at least two kids in class with the same first and family name?

D133

#### DavidSnider

Gold Member
145. What is the smallest natural number with a multiplicative persistence of $5$ in the decimal system?
The multiplicative digit root is the repeated process of multiplying all digits, e.g.
$$3784 \longrightarrow 672 \longrightarrow 84 \longrightarrow 32 \longrightarrow 6$$
Tme multiplicative persistence is the number of arrows, iterations. For $3784$ it is $4$.

D132
Python:
def mp(n,o):
digits = [int(c) for c in str(n)]
prod = 1
for d in digits:
prod = prod * d

if prod < 10:
return o

return mp(prod,o+1)

n = 10
while mp(n,1) < 5:
n+=1

print(n)

#### fresh_42

Mentor
2018 Award
147. In a chess tournament every player has a match with everyone else. The winner will receive a green card, the loser a red one, and in case of a remis, they will receive a yellow card each. At the end of the tournament there have been distributed exactly 752 cards of each color. How many competitors have been in the competition?

D133

#### BvU

Homework Helper
147.
There have been 752 *3 /3 = 2256 matches
2256 = 48 * 47 so there were 49 players who each played 47 matches

#### fresh_42

Mentor
2018 Award
There have been $752 + \frac{1}{2}\cdot 752$ matches.

#### BvU

Homework Helper
Yeah allright. 47*48/2 Keep nitpicking -- we do too

Takes two to tango they say

#### fresh_42

Mentor
2018 Award
Yeah allright. 47*48/2 Keep nitpicking -- we do too

Takes two to tango they say
Yes, but there also only 48 players for 47 matches each.

Well, that was an easy one. The only hurdle was to avoid double counting. With the usual points instead of cards it would have been even easier.

148. 100 kilograms of fruits lie to dry in the sun. The water content is initially at 99 percent. If the proportion is only 98 percent, how heavy are the fruits?

D133

#### BvU

Homework Helper
We're gliding here, aren't we ? This is high school stuff ! 50 kg

#### fresh_42

Mentor
2018 Award
We're gliding here, aren't we ? This is high school stuff ! 50 kg
Yeah, and I even skipped a couple of others being either obvious or solvable by code. But 146. is more challenging and still open! Or the relatively easy 140.

BvU

"Riddles and Puzzles: Extend the following to a valid equation"

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