Can You Solve The Impossible Age And House Number Puzzle?

Thanks to Filippo for the suggestion! This problem is from the 2016 International Championship of Mathematical Games.

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Solution slide 17

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[deerho]

I worked it much more simply. To get the .16 in the average, you had to have 25, 50 or 100 in the denominator of the average. So we were looking at 26, 51, or 101 houses to start. A little trial and error starting with 26 houses quickly got me to the answer. Sum of 1 to 26 is 351. You want something divisible by 25 with a remainder of 4. Subtracting 22 from 351 gives you 329 which fits the bill. 329/4 = 13.16, so that makes Benoit 7.

[David-idjw]

The smallest possible average house number value is 1.5 (1+2, where Benoit lives in house 3). That means Benoit's age has to be less than 18.66, which means his maximum age is 18.

The smallest possible age for Benoit is 1, which means the overall average cannot be created with more than 38 houses, since the average of 39 houses would be 19.5, and the upper limit is 19.16 (20.16 - 1).

The 0.16 fractional leftover has to be evenly created using the number of houses being averaged. 16/100 is not possible because 100 is more than 38 (which we already determined is the maximum number of houses). 8/50 is not possible for the same reason. 4/25 is possible, and it also can't be reduced any further. Thus we have 26 houses on the street, and 25 of those are being averaged together.

The range of possible averages can be found by removing either the 1st or the 26th house, and finding the resulting average. The sum of 1..26 is 351. If you remove the first house, the average is (351-1)/25 = 14.0. If you remove the last house, the average is (351-26)/25 = 13.

Benoit's age thus has to be between (20.16-14) = 6.16 and (20.16-13) = 7.16. The only whole number in that range is 7. Benoit's age is 7.

The house average is thus the remaining 13.16. Multiply by the number of houses in the average (25) and we get 329. Taking the difference between that and the sum of all the houses (351), we get 22, which is Benoit's house number, which wasn't included in the average.

[verkuilb]

Some of your recent problems have been too simple to be deserving of your channel. This one definitely brings you back on course!! Well done!!

[bjorneriksson]

After a bit of thinking, I followed what I now see is the same path as several of the other commenters and realized that 0.16 = 4/25 meant that the number of houses had to be a multiple of 25, and since the average of 50 houses is 25.5 which is higher than the answer of 20.16, the number of houses must be 25, and the sum of the house numbers must also give a remainder of 4 when divided by 25. 1+...+26 = 351, which means that you have to subtract house number 22 to get 329 which is 13 + 4/25. This finally gives you Benoits age of 7.

[benoitlavergne]

Omg, Presh's talking about me ! This riddle sounds very pleasant to my ear, thank you.

Now I'll pause the video, and give this problem a try.

[SG]

I did a little short and less complicated. Since the average of the house numbers is the only way to get a fraction in the results, and 16/100=4/25 we can conclude that the number of houses in the street must have a remainder of 1 when dividing by 25. With 51 the minimum average would be reached with Benoits house number being 51, the average of the remaining houses would be (51*50/2)/50=25.5 which is too large already as Benoit can not have a negative age. For all greater numbers the average would increase even more, therefor we know already that the number of houses must be 26. If Benoits house number is x, the average is
702 has the remainder 2 when dividing by 50. To get the decimal numbers 16 behind the dot we need 702-2x to have the remainder of 8 when dividing by 50. Therefor 2x needs to have the remainder of -6 which equals the remainder of 44. Since x needs to be less or equal to 26 2x needs to be less or equal to 52, the only possibility for 2x is therefor 44, which means x=22. From her we can easily calculate the average house numbers which equals (702-44)/50=13.16. We can now conclude that Benoits age must be 7 years with 20.16-13.16=7.

[FinetalPies]

My solve was so much easier than all that algebra. I just poked around with calculating the case for three or four houses and quickly noticed:
-Removing 1 number from the average can at most change the result by 1
-The amount the average result changes is done in increments of 1/(n-1)

For example for four houses, depending on which house is being removed, the resulting average is 3, 2.67, 2.34, 2. Or in other words, 9/3, 8/3, 7/3, 6/3. Four houses means moving in increments of 1/3.

So, because the end result ends in a .16, or a 4/25, there must be 26 houses to move in increments of 1/25. I did the average of if Benoit was in house 26, which was 13, which is where I probably could've just said that Benoit is 7; but to be sure I calculated the 4th result down from 26 to add the 4/25, and it was 13.16. 7 off from the 20.16 goal.

[duduzilezulu]

I can succesfully say I understood nothing but knew I should stop as soon as you said it was from the international championship of mathematics. I saved my self a headache. Thanks for the tip

[mateuszbok]

I actually figured this out with much less math and some guessing: since the age is a whole number, the .16 in the sum must come from calculating the average. Since 0.16 = 4/25 i guessed that there were 25+1=26 houses. Then the average can be calculated to be 13 + (26-b)/25. If b = 22 then it checks out - the average is 13.16 and in this case the age must be 7.
This method does not check if this is the only answer but it's still nice :)

[sschaubal]

Easier way would have been to identify that the decimal part of the average is unaffected by addition of his age to the average because his age is an integer. Now, 0.16 can only become an integer when multiplied by 25 or some multiple of 25 (like 50). This is important because the number of houses cannot be in decimal and will always be an integer. So, the houses are one more than 25 (like 26) or one more than 50 (like 51) and so on. This is because Benoit's own house is skipped here.
Also, Benoit's age and the integer part of the average add up to 20 (X+Y = 20) as the number comes to 20.16.
Let, X be Benoit's age and Y be the integer (whole number) part of the average.
If we were to multiply (Y +0.16) with 25 (say) as explained earlier, it comes to 25Y + 4 which is equal to 26 X 27 / 2 - N
Here, using the formula n(n+1)/2 for addition of 26 houses in total and the subtracting Benoit's house number N from it
Thus, 25Y + 4 = 351 - N
25Y + N = 347
N now needs to be such that Y is an integer, so it will have to be 22 cause then we have 25Y = 325 and Y comes as 13. We cannot choose N to be 47 as we assumed number of houses to be only 26 here so Benoit's house cannot be numbered 47.
If we were to try with 51 houses next, Y comes to 26, which is not possible as Benoit's age and Y are equal to 20 only.
So, Number of houses is 26 and Benoit's house is numbered 22 and because Y is 13, and X+Y = 20, Benoit is 7 year's old.

[randomguy-tgok]

Commenting before watching the solution:


Today being Benoit's birthday means that his age is not fractional.
The average of all the house numbers except for Benoit's is some fractional number that ends in .16. The question helpfully points out that this is equal to 16/100 which simplifies to 4/25.
So it can be reasonably concluded that there are 26 houses in Benoit's street.
The 25th Triangular number is 325, averaging out to 13. This would be if Benoit lived in house 26. If he lives in 25, the average is 1/25 higher, so if he lives in house 22, the average of every other house's number is 13.16.
This means that Benoit is 7 - a reasonable age for someone to be doing this if you're in a maths challenge. I don't know the quality of the French education system so I don't know if your average French 7-year-old could do this given enough time.

[stanimirF]

The fraction of the sum is .16=4/25. So the number of houses (except his should be multiple of 25). But if it is 50 or more then the average will be 25.5 or greater. That would lead to the age added will be negative. Therefore it is 25. So the total number of houses is 26. The sum of 1 to 26 is 351. But the remainder of the sum divided by 25 should be 5. So his house should be such number that subtracted from 351 would give difference that has 4 as a remainder when divided by 25. Therefore his house is 22 (351-22=329/25=13 4/25 = 13.16). And his age is 20.16-13.16 = 7.

[stevenz]

Just used a simple Excel Visual Basic macro and it solved it in about 1 second 😀 I let n = number of homes and x = Benoit's house number where x <= n. Result was n = 26, x = 22 which resulted in an average of 13.16. Subtracting this from 20.16 equals Benoit's age of 7 👍

[borisbogdanoski]

It can be concluded without long equations and paperwork. Actually, fractional part can be only from the average of house numbers and 16/100 is 4/25. So, denominators can be 25, 50, 75... That means the number of houses are 1 above denominator (as Benoit's house is subtracted). 50 or more houses are not possible (actiually, no more than 40) as average must be less than 20. That implies the number of houses are 26, sum of all is 351 (27*26/2). The smaller number that will give the remainder of 4 when divided by 25 is 329 (Benoit's house is 22) and then the average number is 13 4/25. Finally, his age is 7.

[NaHBrO]

The average of houses is 20.16-x, so the number of houses (n-1) must be a multiple of 25. n=25k+1
if there are n houses, the average of n-1 houses is at least n/2, at most n/2+1 (if the benoit's house number is 1 or n, factorise and calculate)
when k=2, n=51, n/2=25.5. So k=2 is impossible, k=1

k=1, the average must be 13.16, his age is 7.

[kurzackd]

5 (FIVE) lines of code in Python, altho *it could technically be done in 3 (THREE), and **_even LESS_*, albeit with reduced readability and increased clutter:
CODE
for houses in range(4, 40):
for withoutH in range (1, houses):
total = sum(range(1, houses+1))
spec_avg = float(total - withoutH) / (houses - 1)
if str(spec_avg).split('.')[1] == "16": print(houses, withoutH, spec_avg)
CODE
Running this produces *the output: "26 22 13.16"*
Therefore, *the number of houses is 26* . _Benoit lives in house #22_ . *His age is 7* (20.16 - 13.16).
.

[michaellundgren]

I wrote a Python program to calculate his age:
I'm new to Python, so there are probably better methods.

# y is the house number
# x is the number of houses

for y in range (1, 40):
for x in range(1, 40):
sum = 0
count = 0
for i in range(1, x+1):
sum = sum + i
count = count + 1
sum = sum - y
count = count - 1
if count > 0:
average = sum / count
fractional_part = average % 1.0
if fractional_part > 0.159 and fractional_part < 0.161:
age = 20.16 - average
print("Benoit's age is", age)

[gamefacierglitches]

(Just watched the video. My solution is basically the same but more logic, less math.)

This is how I worked it out.

The equation for the sum of numbers 1 to n = n(n+1)/2.

If we were taking the average of all the houses then we would divide by n.

Average = (n+1)/2

At the minimum age of 0, this means taken average is around 20.16 and therefore n=~41 at the very most. The taken average does not include his house thus you would divide by (n-1) instead, but since "41/40=1.025" is a insignificant difference you can still say n = ~41 at most.

Next, a big hint is the taken average leaves a decimal of 0.16 which can be used to identify the denominator. With a small range of 1 to 42, you can divide each number by 0.16 to find an integer (or reduce 16/100). Doing so yields that multiples of 25 can yield 0.16, however due to the restricted range only 25 works.
Thus n-1=25, therefore n=26.

The taken average can be written as:
[total - H]/(n-1)
= total/(n-1) - H/(n-1)
where the total would be n(n+1)/2.

Substituting n=26 we get:
14.04 - H/25

At this point H doesn't matter; only the fact that H is between 1 and 26, and therefore H/25 is between 0.04 and 1.04. The only value with a 0.16 decimal between 13 and 14 is 13.16, and that is the average of the houses.

Simple math solves that the added age is 7yrs old.

At this point the question is a bit confusing with "Today is Benoit's birthday."
- Was 20.16 calculated before today? If so he is 8yrs old.
- Was 20.16 calculated today? If so he is 7yrs old.

Thus Benoit is 8yrs old, or if 20.16 was calculated today then he is 7yrs old.

[PhilipMurphy]

Always good to have education content on YouTube

[vyomsinha]

At the age of 7 Benoit did that calculation.... wow