Puzzles
8 December
It is possible to arrange 4 points on a plane and draw non-intersecting lines
between them to form 3 non-overlapping triangles:
It is not possible to make more than 3 triangles with 4 points.
What is the maximum number of non-overlapping triangles that can be made by arranging 290 points on a plane
and drawing non-intersecting lines between them?
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By adding a point inside a triangle, you can turn one triangle into three triangles. By adding a point outside all the current triangles, you can add either
one or two more triangles. As we are after the maximum number of triangles, we will choose to add two triangles for each point we add.
Four points make 3 triangles. Adding 286 more points will add 572 more triangles, giving a total of 575.
7 December
What is the obtuse angle in degrees between the minute and hour hands of a clock at 08:22?
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Every hour, the hour hand moves 360÷12 = 30°.
Every minute, the hour hand moves 30÷60 = ½° and the minute hand moves 360÷60 = 6°.
At 8:22, the hour hand will be 8×30+22×½ = 251° from the top of the clock
and the minute hand will be 22×6 = 132° from the top of the clock. The angle between the hands will
be 251-132 = 119°.
6 December
The number n has 55 digits. All of its digits are 9.
What is the sum of the digits of n3?
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We can look for a pattern as we increate the number of 9s that make up n:
| n | n3 |
| 9 | 729 |
| 99 | 970299 |
| 999 | 997002999 |
| 999 | 999700029999 |
| 9999 | 999970000299999 |
If n has k digits, then n3 is
k-1 9s,
followed by a 7,
followed by k-1 0s,
followed by a 2,
followed by k 9s,
The sum of all these digits will by 18k.
Hence, the answer is 18×55 = 990.
5 December
The sum of 11 consecutive integers is 2024. What is the smallest of the 11 integers?
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Call the smallest number \(n\). The sum of the 11 integers is:
$$n + (n+1) + (n+2) + \dots + (n+10)$$
This simplifies to:
$$11n +55$$
If \(11n+55=2024\), then \(n\) is 179.
4 December
The geometric mean of a set of n numbers is computed by mulitplying all the
numbers together, then taking the nth root.
The factors of 9 are 1, 3, and 9. The geometric mean of these factors is
$$\sqrt[3]{1\times3\times9}=\sqrt[3]{27}=3$$
What is the smallest number where the geometric mean of its factors is 13?
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Square numbers are the only numbers where the geometric mean of their factors is equal to a whole number,
and in each case the geometric mean will be the square root. Therefore the only number
where the geometric mean of its factors is 13 is 169.
3 December
There are 5 ways to write 5 as the sum of positive odd numbers:
- 1 + 1 + 1 + 1 + 1
- 1 + 1 + 3
- 3 + 1 + 1
- 1 + 3 + 1
- 5
How many ways are there to write 14 as the sum of positive odd numbers?
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This can be solved by working it out for some examples then looking for the pattern:
| Total | Ways to make | Number of ways |
| 1 | 1 | 1 |
| 2 | 1+1 | 1 |
| 3 | 1+1+1, 3 | 2 |
| 4 | 1+1+1+1, 3+1, 1+3 | 3 |
| 5 | 1+1+1+1+1, 1+1+3, 1+3+1, 3+1+1 | 5 |
| 6 | 1+1+1+1+1+1, 1+1+1+3, 1+1+3+1, 1+3+1+1, 3+1+1+1, 3+3, 5+1, 1+5 | 8 |
The looks like the Fibonacci numbers: every term is the sum of the previous two terms.
Continuing the pattern gives 377 ways to make 14.
To justify why the answer is the Fibonacci numbers, notice that you split the sums for a number n into two sets:
those that end with "+1" and those that end with something else.
Those that end with "+1" are a way of making n-1, plus the one on the end.
Those that don't end with "+1" are a way of making n-2, with two added to the final number.
So the number of ways of making n is
the number of ways of making n-1 plus the number of ways of making n-2.
2 December
14 is the smallest even number that cannot be obtained by rolling two 6-sided dice and finding
the product of the numbers rolled.
What is the smallest even number that cannot be obtained by rolling one hundred 100-sided dice
and finding the product of the numbers rolled?
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With two 100-sided dice, it is possible to roll the even numbers from 2 to 200 by rolling a two on one die
and 1 to 100 on the other. It is not possible to roll 202, as 101 is prime so cannot be made by multiplying any numbers less than 101.
With one hundred 100-sided dice, the even numbers from 2 to 200 can still be made (by rolling the two dice as before plus lots of ones).
It is still not possible to roll 202, as 101 is still prime.
1 December
Eve writes down five different positive integers. The sum of her integers is 16. What is
the product of her integers?
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The only five different positive integers with a sum of 16 are 1, 2, 3, 4, and 6.
The product of these is 144.
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