### Brain Teasers

# 861 Candles

I have always had the correct number of candles on each of my birthday cakes. I have had exactly one birthday cake for each year of my life. If I have blown out a total of 861 candles thus far, how old am I?

Can you find the answer without just adding up numbers until you reach 861?

Can you find the answer without just adding up numbers until you reach 861?

### Hint

1+ 2+ 3 + ... + n = 861 .To sum up the first n positive integers, the following formula is helpful:

Sum = n*(n+1)/2

Think of it this way: find the smaller of two consecutive positive integers such that half the product of the integers is 861.

### Answer

41If n = 41 , we see that:

S = 41(41+1)/2 = 41(42)/2 = 861.

Thus, if you add up the integers from 1 to 41, you will get a total of 861.

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## Comments

Im surprised the difficult on this one isn't higher. You must solve a quadratic equation to get the answer. Does anyone actually rememer their algebra? Not I. Great teaser.

You don't need to remember algebra, or to have ever studied it.

You just need to remember some arithmetic (you don't even need a calculator):

You reason: adding all integers from 1 to whatever give the same result as adding them from whatever down to 1.

Think of adding the two sums line by line. On the first line you add 1 and whatever. On the second line, 2 and one less than whatever, ...., and on the last line whatever plus 1. You end up with whatever lines each equal to whatever plus 1. All together that is whatever times 1 more than whatever. And that is twice the sum of the candles, 861, which is 1722.

Now you are going to try numbers. 17 is a little more than 16, and you do remember that 4x4=16. So 40x40 is 1600 and 40x41 is 1600+40=1640. That is not quite enough. Perhaps 41x42 will work. But you don't remember how to multiply two-digit numbers.

But you know how to multiply by 2.

2x41 is 82. And you remember how to add 1640+82=1722. That's the number we wanted!

So "whatever" is 41.

What I am not so sure about is: did he really blow the candle out himself when he was 1?

You just need to remember some arithmetic (you don't even need a calculator):

You reason: adding all integers from 1 to whatever give the same result as adding them from whatever down to 1.

Think of adding the two sums line by line. On the first line you add 1 and whatever. On the second line, 2 and one less than whatever, ...., and on the last line whatever plus 1. You end up with whatever lines each equal to whatever plus 1. All together that is whatever times 1 more than whatever. And that is twice the sum of the candles, 861, which is 1722.

Now you are going to try numbers. 17 is a little more than 16, and you do remember that 4x4=16. So 40x40 is 1600 and 40x41 is 1600+40=1640. That is not quite enough. Perhaps 41x42 will work. But you don't remember how to multiply two-digit numbers.

But you know how to multiply by 2.

2x41 is 82. And you remember how to add 1640+82=1722. That's the number we wanted!

So "whatever" is 41.

What I am not so sure about is: did he really blow the candle out himself when he was 1?

It only took about five seconds to get it. All you do is double the number and take the square root. Assuming there is an answer, it will be the integer portion.

Since I've memorized all the squares up to 50+, I immediately knew that 1722 was between 41^2 = 1681 and 42^2 = 1764.

Since I've memorized all the squares up to 50+, I immediately knew that 1722 was between 41^2 = 1681 and 42^2 = 1764.

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