Mathematical Problems

They both start off at the same time towards the opposite bank. They pass each other at 180 meters from the bank where Patrick departed. When reaching the opposite bank, they both take a rest for the same amount of time before they return. On the way back they pass each other at 100 meters from the bank from where Patrick returned.

Patrick and Eric both row with a constant speed, but Eric rows faster.

• all digits from 0 up to 9 occur exactly once in the number;
• the first digit is divisible by 1;
• the number formed by the first two digits is divisible by 2;
• the number formed by the first three digits is divisible by 3;
• the number formed by the first four digits is divisible by 4;
• the number formed by the first five digits is divisible by 5;
• the number formed by the first six digits is divisible by 6;
• the number formed by the first seven digits is divisible by 7;
• the number formed by the first eight digits is divisible by 8;
• the number formed by the first nine digits is divisible by 9;
• the number formed by the ten digits is divisible by 10.

She asks him: "Is your number larger than 50?"
William answers, but lies.
Upon this, Lisa asks: "Is your number a multiple of 4?"
William answers, but lies again.
Then Lisa asks: "Is your number a square?"
William answers truthfully.
Upon this, Lisa says: "I know your number if you tell me whether the first digit is a 3."
William answers, but now we don't know whether he lies or speaks the truth.

Thereupon, Lisa says at which number she thinks William lives, but (of course) she is wrong.

123456789 = 100

In a certain week, Pat did not deliver any mail at two houses in the village; at the other houses he delivered mail three times each. Each working day he delivered mail at exactly four houses.

The sums of the house numbers where he delivered mail were:
on Monday: 18
on Tuesday: 12
on Wednesday: 23
on Thursday: 19
on Friday: 32
op Saturday: 25
on Sunday: he never works

Copyright © 2002 E.R. van Veldhoven. All rights reserved.

We discovered a curious thing. We took the number 461, reversed the digits, giving the number 164, and calculated the sum of these two numbers:

       461
       164 +
     -------
       625

We repeated the process of reversing the digits and calculating the sum two more times:

       625
       526 +
     -------
      1151
      1511 +
     -------
      2662

To our surprise, the result 2662 was a palindrome. We decided to see if this was a pure coincidence or not. So we took another 3-digit number, reversed it, which gave a larger number, and added the two. The result was not a palindrome.

We repeated the process, which resulted in another 3-digit number which was still not a palindrome. We had to repeat the process twice more to finally arrive at a 4-digit number which was a palindrome.