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;
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?
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?
15 December
The arithmetic mean of a set of \(n\) numbers is computed by adding up all the numbers, then
dividing the result by \(n\).
The geometric mean of a set of \(n\) numbers is computed by multiplying all the numbers together, then
taking the \(n\)th root of the result.
The arithmetic mean of the digits of the number 132 is \(\tfrac13(1+3+2)=2\).
The geometric mean of the digits of the number 139 is \(\sqrt[3]{1\times3\times9}\)=3.
What is the smallest three-digit number whose first digit is 4 and for which the arithmetic and geometric means of its digits are both non-zero integers?
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The only three digit number whose first digits is 4 and for which the arithmetic and geometric means of its digits are both non-zero integers is 444.
Extension
How many three digit numbers are there for which the arithmetic and geometric means of its digits are both non-zero integers?
How many four digit numbers are there for which the arithmetic and geometric means of its digits are both non-zero integers?
12 December
What is the smallest value of \(n\) such that
$$\frac{500!\times499!\times498!\times\dots\times1!}{n!}$$
is a square number?
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Consider the first two terms in the product:
$$
500!\times499!
=500\times499!\times499!$$
$$= 500\times(499!)^2.$$
Doing similar steps with each pair of terms in the product, we see that:
$$
500!\times499!\times498!\times\dots\times1!
=
500\times498\times\dots\times2\times(499!\times497!\times\dots\times1!)^2
$$
$$
=
(2\times250)\times(2\times249)\times\dots\times(2\times1)\times(499!\times497!\times\dots\times1!)^2
$$
$$
=
2^{250}\times250!\times(499!\times497!\times\dots\times1!)^2
$$
If we divide this by \(250!\), we are left with a square number, and so \(n\) is 250
11 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.
| + | | + | | = 15 |
| + | | + | | ÷ | |
| + | | – | | = 10 |
| + | | – | | × | |
| ÷ | | × | | = 3 |
=
16 | | =
1 | | =
30 | |
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| 3 | + | 7 | + | 5 | = 15 |
| + | | + | | ÷ | |
| 9 | + | 2 | – | 1 | = 10 |
| + | | – | | × | |
| 4 | ÷ | 8 | × | 6 | = 3 |
=
16 | | =
1 | | =
30 | |
The product of the numbers in the red boxes is 120.
10 December
How many integers are there between 100 and 1000 whose digits add up to an even number?
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Between 100 and 109 (inclusive), there are 5 integers whose digits add up to an even number, and 5 whose digits add up to an odd number.
Between 110 and 119 (inclusive), there are 5 integers whose digits add up to an even number, and 5 whose digits add up to an odd number...
In general, between \(10n\) and \(10n+9\) (inclusive), there are 5 integers whose digits add up to an even number, and 5 whose digits add up to an odd number.
The integers from 100 to 999 (inclusive) can be split into 45 sets of integers from \(10n\) to \(10n+9\) (and the digits of 1000 don't add to an even number), so there are
450 integers between 100 and 1000 whose digits add up to an even number.
