Puzzles
24 December
When written in binary, the number 235 is 11101011. This binary representation starts and
ends with 1 and does not contain two 0s in a row.
What is the smallest three-digit number whose binary representation starts and
ends with 1 and does not contain two 0s in a row?
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- 100 in binary is 1100100;
- 101 in binary is 1100101;
- 102 in binary is 1100110;
- 103 in binary is 1100111;
- 104 in binary is 1101000;
- 105 in binary is 1101001;
- 106 in binary is 1101010;
- 107 in binary is 1101011;
23 December
There are 18 ways to split a 3 by 3 square into 3 rectangles whose sides all have integer length:
How many ways are there to split a 10 by 10 square into 3 rectangles whose sides all have integer length?
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The square is split into 3 rectangles by drawing two lines on the rectangle. There are two cases: the lines can both go
in the same direction; or the lines go in perpendicular directions, with one going all the way across the square and one going from an edge of the square to the other line.
For an \(n\) by \(n\) square, there are:
- \((n-1)(n-2)/2\) ways to pick two dividing lines that are both vertical;
- \((n-1)(n-2)/2\) ways to pick two dividing lines that are both horizontal;
- \(2(n-1)(n-1)\) ways to pick two dividing lines where one is vertical and goes all the way across the square, and the other is horizontal.
- \(2(n-1)(n-1)\) ways to pick two dividing lines where one is horizontal and goes all the way across the square, and the other is vertical;
In total this makes \((n-1)(5n-6)\) ways to split the square. (10–1)×(5×10-6) is 396.
22 December
There are 4 ways to pick three vertices of a regular quadrilateral so that they form a right-angled triangle:
In another regular polygon with \(n\) sides, there are 14620 ways to pick three vertices so that they form a right-angled triangle. What is \(n\)?
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The vertices of any regular polygon lie on a circle, and so the three vertices we pick must lie on this circle. The triangle will be right-angled
if one of the edges of the triangle is a diameter of the circle.
If the regular polygon has \(n=2k\) sides, then there are \(k\) ways to pick two points that make a diameter, and \(2k-2\) ways to pick the third point to make the triangle, and so there are
a total of \(k(2k-2)=n(n-2)/2\) right-angled triangles. This means that we need to solve \(n(n-2)/2=14620\). The solution is \(n=172\), and so our polygon has 172 sides.
21 December
There are 6 two-digit numbers whose digits are all 1, 2, or 3 and whose second digit onwards
are all less than or equal to the previous digit:
How many 20-digit numbers are there whose digits are all 1, 2, or 3 and whose second digit onwards
are all less than or equal to the previous digit?
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We can look at how many \(n\)-digit number there are for small values of \(n\) and look for a pattern:
- 1-digit numbers: there are 3.
- 2-digit numbers: there are 6.
- 3-digit numbers: there are 10.
- 4-digit numbers: there are 15.
These are the triangle numbers, and there are 231 20-digit numbers.
Extension
Why is the pattern the triangle numbers?
20 December
There are 6 different ways that three balls labelled 1 to 3 can be put into two boxes labelled
A and B so that no box is empty:
How many ways can five balls labelled 1 to 5 be put into four boxes labelled A to D so that
no box is empty?
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One method for putting the five balls into the four boxes is:
pick one of the balls (there are 5 ways to do this), then put the other four balls into boxes with one in each box (there are 4! ways to do this), then
put the picked ball into a box (there are 4 ways to do this). This method gives exactly two ways of obtaining each possible arrangement, so the number of
ways is \(5\times4!\times4\div2\) or 240.
19 December
Put the digits 1 to 9 (using each digit exactly once) in the boxes so that the sums are correct. The sums should be read left to right and top to bottom ignoring the usual order of operations. For example, 4+3×2 is 14, not 10.
Today's number is the product of the numbers in the red boxes.
| + | | – | | = 7 |
| × | | × | | × | |
| + | | – | | = 0 |
| ÷ | | ÷ | | ÷ | |
| + | | – | | = 2 |
=
4 | | =
35 | | =
18 | |
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| 8 | + | 5 | – | 6 | = 7 |
| × | | × | | × | |
| 2 | + | 7 | – | 9 | = 0 |
| ÷ | | ÷ | | ÷ | |
| 4 | + | 1 | – | 3 | = 2 |
=
4 | | =
35 | | =
18 | |
The product of the numbers in the red boxes is 504.
18 December
Some numbers can be written as the product of two or more consecutive integers, for example:
$$6=2\times3$$
$$840=4\times5\times6\times7$$
What is the smallest three-digit number that can be written as the product of two or more consecutive integers?
17 December
If you expand \((a+b+c)^2\), you get \(a^2+b^2+c^2+2ab+2ac+2bc\).
This has 6 terms.
How many terms does the expansion of \((a+b+c+d+e+f)^5\) have?
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