The legendary king Midas
possessed a huge amount of gold. He hid this treasure carefully: in a building
consisting of a number of rooms. In each room there were a number of boxes; this
number was equal to the number of rooms in the building. Each box contained a
number of golden coins that equaled the number of boxes per room. When the king
died, one box was given to the royal barber. The remainder of the coins had to
be divided fairly between his six sons. Is a fair division possible in all
situations?
A fair division of Midas' coins is indeed possible. Let the number of rooms be
N. This means that per room there are N boxes with N coins each. In total there
are N×N×N = N3 coins. One box with N coins goes to the barber. For the
six brothers, N3 - N coins remain. We can write this as: N(N2 - l), or: N(N -
1)(N + l). This last expression is divisible by 6 in all cases, since a number
is divisible by 6 when it is both divisible by 3 and even. This is indeed the
case here: whatever N may be, the expression N(N - 1)(N + l) always contains
three successive numbers. One of those is always divisible by 3, and at least
one of the others is even. This even holds when N=1; in that case all the
brothers get nothing, which is also a fair division!
On a sunny morning, a
greengrocer places 200 kilograms of cucumbers in cases in front of his shop. At
that moment, the cucumbers are 99% water. In the afternoon, it turns out that it
is the hottest day of the year, and as a result, the cucumbers dry out a little
bit. At the end of the day, the greengrocer has not sold a single cucumber, and
the cucumbers are only 98% water. How many kilograms of cucumbers has the
greengrocer left at the end of the day?
In the morning, the 200 kilograms of cucumbers are 99% water. So the non-water
part of the cucumbers has a mass of 2 kilograms. At the end of the day, the
cucumbers are 98% water. The remaining 2% is still the 2 kilograms of non-water
material (which does not change when the water evaporates). If 2% equals 2
kilograms, then 100% equals 100 kilograms. So, the greengrocer has 100 kilograms
of cucumbers left at the end of the day.
A swimmer jumps from a
bridge over a canal and swims 1 kilometer stream up. After that first kilometer,
he passes a floating cork. He continues swimming for half an hour and then turns
around and swims back to the bridge. The swimmer and the cork arrive at the
bridge at the same time. The swimmer has been swimming with constant speed. How
fast does the water in the canal flow?
If you have written down a full paper of mathematical formulas, you have been
thinking too complicated...It is obvious that the cork does not move relatively
to the water (i.e. has the same speed as the water). So if the swimmer is
swimming away from the cork for half an hour (up stream), it will take him
another half hour to swim back to the cork again. Because the swimmer is
swimming with constant speed (constant relatively to the speed of the water!)
you can look at it as if the water in the river doesn't move, the cork doesn't
move, and the swimmer swims a certain time away from the cork and then back. So
in that one hour time, the cork has floated from 1 kilometer up stream to the
bridge. Conclusion: The water in the canal flows at a speed of 1 km/h..
Consider a road with two
cars, at a distance of 100 kilometers, driving towards each other. The left car
drives at a speed of forty kilometers per hour and the right car at a speed of
sixty kilometers per hour. A bird starts at the same location as the right car
and flies at a speed of 80 kilometers per hour. When it reaches the left car it
turns its direction, and when it reaches the right car it turns its direction
again to the opposite, etcetera. What is the total distance that the bird has
traveled at the moment that the two cars have reached each other?
If you have written down a full paper of mathematical formulas, you haven't been
thinking in the right direction. It is obvious that the two cars meet each other
after one hour. On that moment, the bird has flown for one hour. Conclusion: The
bird has flown 80 km/h × 1 h = 80 km. .
On a sunny morning, a
greengrocer places 200 kilograms of cucumbers in cases in front of his shop. At
that moment, the cucumbers are 99% water. In the afternoon, it turns out that it
is the hottest day of the year, and as a result, the cucumbers dry out a little
bit. At the end of the day, the greengrocer has not sold a single cucumber, and
the cucumbers are only 98% water. How many kilograms of cucumbers has the
greengrocer left at the end of the day?
In the morning, the 200 kilograms of cucumbers are 99% water. So the non-water
part of the cucumbers has a mass of 2 kilograms. At the end of the day, the
cucumbers are 98% water. The remaining 2% is still the 2 kilograms of non-water
material (which does not change when the water evaporates). If 2% equals 2
kilograms, then 100% equals 100 kilograms. So, the greengrocer has 100 kilograms
of cucumbers left at the end of the day..
A number is called a
palindrome when it is equal to the number you get when all its digits Postman
Pat delivers the mail in the small village Tenhouses. This village, as you
already suspected, has only one street with exactly ten houses, numbered from 1
up to and including 10. 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 Which
two houses didn't get any mail that week?
If postman Pat would have delivered mail three times at each house, then the
total sum of the house numbers per day would be
(1+2+3+4+5+6+7+8+9+10)×3=165. Now that sum is 18+12+23+19+32+25=129. The
difference is 165-129=36; divided by 3 this is 12. The sum of the house numbers
where no mail was delivered is therefore 12. The following combinations are
possible: 2+10
3+9
4+8
5+7
Each day at four houses the mail was delivered. On Tuesday the sum was 12. 12
can only be made from four house numbers in 2 ways:
1+2+3+6
1+2+4+5
The same holds for Friday with the sum of 32
5+8+9+10
6+7+9+10
From this we can conclude that the house numbers 1, 2, 9 and 10 for sure have
received mail, which means that the combinations 2+10 and 3+9 are not possible.
Also the combination 5+7 is not possible, because mail was delivered either at
house 5 or at house 7. Thus the only remaining solution is: houses 4 and 8.
N.B.: there are various possibilities for the actual post delivery of the whole
week. For example: Monday houses 1, 3, 5 and 9
Tuesday houses 1, 2, 3 and 6
Wednesday houses 1, 5, 7 and 10
Thursday houses 2, 3, 5 and 9
Friday houses 6, 7, 9 and 10
Saturday houses 2, 6, 7 and 10 .
You walk upwards on an
escalator, with a speed of 1 step per second. After 50 steps you are at the end.
You turn around and run downwards with a speed of 5 steps per second. After 125
steps you are back at the beginning of the escalator. How many steps do you need
if the escalator stands still?
Let v be the speed of the escalator, in steps per second. Let L be the number of
steps that you need to take when the escalator stands still. Upwards (along with
the escalator), you walk 1 step per second. You need 50 steps, so that takes 50
seconds. This gives: L - 50 × v = 50. Downwards (against the direction of
the escalator), you walk 5 steps per second. You need 125 steps, so that takes
25 seconds. This gives: L + 25 × v = 125. From the two equations follows: L
= 100, v = 1. When the escalator stands still, you need 100 steps..
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