S

If I called out the numbers between 1 and 100 in random order, skipping one, how would you determine the missing number without the assistance of a pencil and paper?

6 Answers
You must do it in your head. I know there are at least two ways it can be done.
T

The nice thing about a question like this is that there is no context for the pauses between the call out of the numbers. If the pauses are too quick it is impossible to figure it out because one would have to have the capacity to remember the numbers to some degree to ascertain a solution. Ergo a reasonable means of receiving the numbers would require at least enough time for the numbers to be uttered and heard.

So the first method is that I would listen and wait and just keep a big checklist in my mind and each time one of the numbers of one through one hundred is called out I would put a check next to the list. Unless the numbers were called out faster than I could process, this would be one of the methods for me to determine the missing number without the assistance of a pencil or paper.

I would just wait and listen. Whichever number is not said is the one, and the gift you've given those who participate in the way the condition works is that there is only one number that will not be said. That means that ninety nine numbers will be said, since the knowledge exists that only one is not, and so I would bide my time. I would keep an extensive list in my mind of the numbers and also maintain a running count in my head for how many numbers have been said. One of the real skills I've developed throughout this life is the capacity to actively listen and process multiple thoughts in my head at the same time.

This skill developed when I was a debater and I had to listen, write notes, and then formulate arguments in response in a very brief period of time if I ever expected to win a debate round. Thousands of debate rounds later that I have debated in or that I have been the judge of now that I am older, I have a very keen ear and a quick mind.

The other way, and this one all depends on how long the pauses are between each number being called out, is that after every number said I would repeat it out loud. This would force me to think about the numbers I've repeated of what was said and then think about the number that was not said because I would be maintaining an ongoing list in my head to reinforce what was said before. Repetition is really key to remembering something, and so if there was a long enough pause between the numbers being called out I could formulate a rhyme or a song to remember all the numbers said. I draw inspiration from the spoken traditions of old that would recount history and myth through spoken tales and stories.

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R

Take each number provided and subtract 50 from it, then keep a running sum of those numbers [(60-50) + (7-50) + (30-50)....]. Half will be negative, half will be positive and unless the set of numbers includes 0, the total sum should end up being -50. If a number is missing from the set, the total will be greater than -50 by a certain amount. Take that number (let's say you end up with -40, so the difference is +10) and subtract it from 100 to get 90, which is the missing number.

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M

The sum of 1+2+3+4+5....+99+100=5050. As you randomly called out the numbers, I could add the figures in my head without pencil and paper. The sum of the numbers you randomly called out would then be subtracted from 5050. The difference would be the number you omitted.

Conversely, I suppose I could subtract the random numbers from 5050 as you called them out, (but adding the numbers and subtracting the sum from 5050 would be easier for me).

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S

Or you could start with 5050 and subtract the numbers as they are called out. Then at the end you subtract the number 1 for the answer. By the way, you could use a calculator as this is not forbidden in the question.

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R

Yet another way is to make all odd numbers - and all even number +. Keep a running sum like before and the answer should be, again, -50. Whatever the difference is from -50 is your missing number.

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S

Very good, Maidti83 There's another method, too. I think it's easier because the numbers remain smaller (hint). Can anyone figure out the second approach?

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