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A woman is sitting in her hotel room when there is a knock at the door. She opened the door to see a man whom she had never seen before. He said “oh I’m sorry, I have made a mistake, I thought this was my room.” He then went down the corridor and in the elevator. The woman went back into her room and phoned security. What made the woman so suspicious of the man?
Answer: You don't knock on your own hotel room door.
Solution:
When the celebrated German mathematician Karl Friedrich Gauss (1777-1855) was nine he was asked to add all the integers from 1 through 100. He quickly added 1 to 100, 2 to 99, and so on for 50 pairs of numbers each adding to 101. Answer: 50 X 101=5,050. What is the sum of all the digits in integers from 1 through 1,000,000,000? (That’s all the digits in all the numbers, not all the numbers themselves.)
Answer: The numbers can be grouped by pairs: 999,999,999 and 0; 999,999,998 and 1' 999,999,997 and 2; and so on.... There are half a billion pairs, and the sum of the digits in each pair is 81. The digits in the unpaired number, 1,000,000,000, add to 1. Then: (500,000,000 X 81) + 1= 40,500,000,001.
Solution:
Show Answer
The numbers can be grouped by pairs:
999,999,999 and 0;
999,999,998 and 1′
999,999,997 and 2;
and so on….
There are half a billion pairs, and the sum of the digits in each pair is 81. The digits in the unpaired number, 1,000,000,000, add to 1. Then:
(500,000,000 X 81) + 1= 40,500,000,001.
Show Answer
The numbers can be grouped by pairs:
999,999,999 and 0;
999,999,998 and 1′
999,999,997 and 2;
and so on….
There are half a billion pairs, and the sum of the digits in each pair is 81. The digits in the unpaired number, 1,000,000,000, add to 1. Then:
(500,000,000 X 81) + 1= 40,500,000,001.
999,999,999 and 0;
999,999,998 and 1′
999,999,997 and 2;
and so on….
There are half a billion pairs, and the sum of the digits in each pair is 81. The digits in the unpaired number, 1,000,000,000, add to 1. Then:
(500,000,000 X 81) + 1= 40,500,000,001.
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