Puzzles
4 December
Some numbers can be written as the sum of four consecutive numbers, for example: 142 = 34 + 35 + 36 + 37.
What is the mean of all the three-digit numbers that can be written as the sum of four consecutive numbers?
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The sum of four consecutive numbers starting at \(n\) is \(n + (n + 1) + (n + 2) + (n + 3) = 4n+6=4(n+1)+2\). This tells us that the numbers
that are the sum of four consecutive numbers are the numbers that are two more than multiples of 4. The smallest three digit number of this form is 102 and the largest is 998: the mean of these
two is 550. The second smallest and second largest are 106 and 994: the mean of these is also 550. All the numbers can be paired up in this way except for 550 itself, and so the mean of all the
numbers is 550.
Extension
What is the mean of all 3-digit numbers that can be written as the sum of \(k\) consective numbers, for different values of \(k\)? When is the answer an integer and when is it not?
24 December
There are 343 three-digit numbers whose digits are all 1, 2, 3, 4, 5, 6, or 7. What is the
mean of all these numbers?
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The digits 1 to 7 will each appear the same number of times as each other in each position of the number, so each digit of the mean will be the mean of the digits 1 to 7.
Therefore the mean is 444.
22 December
22 is two times an odd number. Today's number is the mean of all the answers on days (including today) that are two times an odd number.
Clarification: You are taking the mean for answers on days that are two times an odd numbers; ie. the days are two times odd, not the answers.
16 December
Today's number is four thirds of the average (mean) of the answers for 13th, 14th, 15th and 16th December.
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Let today's number be \(n\). The average is:
$$\frac{729+313+328+n}{4}$$
Therefore:
$$\begin{array}{rl}
\frac34n&=\frac{729+313+328+n}{4}\\
3n&=729+313+328+n\\
2n&=729+313+328\\
&=1370\\
n&=685
\end{array}$$
