Number Riddles

Marian Small

n attractive part of math for many students is solving puzzles or riddles. Students at almost all ages simply enjoy figuring out a puzzle. When teachers provide students with appropriate riddles for their age group and encourage students to create their own riddles, those students are likely to focus on and use the important properties of numbers and operations that we want them to revisit. It is almost an ideal way to get them to practise math happily.

Let’s look at possible riddles for different grade levels.

Riddles for Grades K–2

These number riddles will cover topics including numeral writing, comparing numbers, number representations, skip counting, decomposition, addition, and subtraction. They will also involve lots of reasoning.

Some examples:

I’m thinking of a number.
When you write it, you see round parts and you also see straight parts.
My number is greater than 4.
What could it be?

Notice that possible answers include 5 and 9, allowing even quite young children to solve this riddle, but there are many more possibilities to challenge students, such as 70 or 100.

I’m thinking of a less than 10.
But on a number path, it takes me more steps to get to 10 than to go back to start.
What could the number be?

There are many possible solutions here. In fact, any number less than 5 works.

I’m thinking of a number on a 100-chart.
It is closer to the left edge of the chart than the right edge.
It is closer to the top of the chart than the bottom.
When you add the number’s digits, the total is less than 6.
What could it be?

Again, there are many possibilities, for example:
11, 12, 13, 14, 21, 22, 23, 31, 32, 41.

I’m thinking of a number.
When you show that value using 10-frames, you use more than 3 whole frames, but fewer than 5 whole frames.
You say the number when you skip count by 5.
What could it be?

There are several possible answers, for example, 35, 40, and 45.

I’m thinking of a number.
You can split it up into three smaller numbers: one of these numbers is the same as two copies of another one of the numbers.
If the number is less than 30, what could it be?

There are many answers here.
Students might make their parts 1 and 2 and anything else up to 26.
Or they might make their parts 5 and 10 and anything else up to 14.
Or they might make their parts 8 and 16 and anything else up to 5, etc.

I’m thinking of a number greater than 10.
When you subtract 10 from my number, you get less than if you subtract my number from 30.
What could the number be?

Notice that any teen number works, since if you subtract 10 from a teen number, you get a single-digit number, but you get a two-digit number if you subtract any teen number from 30.
But then a student might wonder if numbers in the 20s work and try those.
In fact, they don’t. 20 does not work since 20 – 10 = 30 – 20 and is not less. The answers are equal. And if you choose a number greater than 20, the first answer gets greater and the second one less. No number in the 20s will work.

Creating Their Own Riddles

You might encourage students to create their own riddles based on representations of numbers, comparisons, ways to decompose numbers, or addition or subtraction.

You might even suggest styles of clues, for example:
When you write the number …
When you add it to …, then …
When you show it on a 10-frame, …
When you compare it to …, then …

Riddles for Grades 3–5

Similar styles of riddles might be proposed for students at these levels, except that the topics addressed would be more appropriate for their grade levels.

Some examples:

I’m thinking of a number that has three digits.
One of the digits is 4 greater than another.
One of the digits is 7 less than another.
What could the number be?

Students might reason in this way: If one digit is 7 less than another, then there must be a 0 and 7 or 1 and 8 or 2 and 9 in the number.
If there were a 0 and 7, the other digit could be 4, so the number could be 407 or 704 or 470 or 740. The other digit could also be 3, so the number could be 307 or 703 or 370 or 730.
If there were a 1 and 8, the other digit could be 5, so the number could be 518 or 581 or 158 or 185 or 815 or 851. The other digit could also be 4, so the number could be 418 or 481 or 148 or 184 or 814 or 841.
If there were a 2 and 9, the other digit could be 6, so the number could be 269 or 296 or 629 or 692 or 926 or 962. The other digit could also be 5, so the number could be 259 or 295 or 529 or 592 or 925 or 952.

I’m thinking of a whole number that is more than 1234 but less than 3142.
When you show it with base ten blocks, you might use 36 blocks.
What could the number be?

Again, there are many possibilities, and so the potential is there to encourage students who are ready to determine all the possibilities.

Since the thousands digit must be 1 or 2 or 3, a student might wonder if the sum of the other digits could be 33 or 34 or 35, but that is not possible with three digits.

So, the student might think of showing 1000 as 10 hundreds or 2000 as 20 hundreds or 3000 as 30 hundreds.

If there were 30 hundreds, the sum of the other digits could be 6, so the number could be 3006 shown as 30 flats and 6 ones or maybe 3060 as 30 flats and 6 tens.

If there were 20 hundreds, the sum of the other digits could be 16, so the number could be 2088 shown as 20 flats, 8 tens, and 8 ones, or it could be 2466 shown as 24 flats, 6 tens, and 6 ones.

Ift there were 10 hundreds, the sum of the other digits could be 26, so the number could be 1998, for example, shown as 19 flats, 9 tens, and 8 ones.
There are other possibilities as well.

I’m thinking of a whole number that you can add to 44 123.
The sum has both a 2 and a 9 in it.
What could the number be?

Again, there are choices.
For example, if you add 106, the sum would be 44 229, so there would be a 2 and 9 in it.

You could also add 206, 306, 406, etc.

But there are other choices too, for example, if you add 8371, 8471, 8571, etc.

I’m thinking of a fraction.
The fraction is closer to 1 than it is to \(\frac{9}{10}\).
But its numerator and denominator are more than 30 apart.
What might the fraction be?

Students are likely to begin by using fractions like \(\frac{11}{12}\) or \(\frac{19}{20}\), but the numerator and denominator are too close.

Eventually they might realize that you need a much greater denominator to make it work.
For example, the fraction \(\frac{260}{300}\) is close to, but a little less than, \(\frac{270}{300}\), which is \(\frac{9}{10}\).
So, maybe it would be better to use \(\frac{365}{400}\), since \(\frac{360}{400}\) is \(\frac{9}{10}\).

I’m thinking of a fraction.
When you write fractions that are equivalent to this one, there are two equivalents with denominators in the 30s, but only one equivalent with a denominator in the 50s.
You cannot simplify the fraction you are thinking of to an equivalent fraction with a smaller denominator.
What might the fraction be?

Students must realize that if the original denominator is small, for example, 2 or 3 or 4, there are several equivalent fractions in the 50s. For example, for \(\frac{1}{2}\), there are equivalents like \(\frac{25}{50}\), \(\frac{26}{52}\), \(\frac{27}{54}\), etc. For \(\frac{1}{3}\), there are equivalents like \(\frac{17}{51}\), \(\frac{18}{54}\), and \(\frac{19}{57}\). For \(\frac{1}{4}\), there are equivalents of \(\frac{13}{52}\) and \(\frac{14}{56}\). So, the denominator must be more than 4.

You might try something like \(\frac{4}{5}\). In the 30s, there would be equivalents with denominators of 30 and 35 (since \(\frac{24}{30}\) and \(\frac{28}{35}\) are equivalent), but there would also be equivalent fractions with denominators of 50 and 55 (\(\frac{40}{50}\) and \(\frac{44}{55}\)). That means the original denominator must be more than 5.

If you try something like \(\frac{5}{6}\), in the 30s, there would be equivalent fractions with denominators of 30 and 36 (\(\frac{25}{30}\) and \(\frac{30}{36}\)), and in the 50s, there would be only one equivalent fraction, with a denominator of 54 (\(\frac{45}{54}\)), so that works.

Students might even try denominators greater than 6. They start to realize that once the denominator gets too big, there will not be two equivalents with denominators in the 30s.

I’m thinking of a decimal.
When you subtract it from 25.13, you end up with digits of 3 and 7 in the difference.
What might the decimal be?

There are many possibilities.
It could be 21.41, since 25.13 – 21.41 = 3.72.
It could also be 25.13 – 11.41, which is 13.72, or 25.13 – 1.41, which is 23.72.

Another choice is 14.76, since 25.13 – 14.76 = 10.37.
You could also change 14.76 to 13.76 or 12.76 or 11.76.

Creating Their Own Riddles

You might encourage students to create their own riddles based on representations of numbers, comparisons, one of the four operations, or fraction or decimal equivalents.

As with primary students, you might even suggest styles of clues, for example:
When you write the number …
When you add it to …, then …
When you show it with base ten blocks, …
When you compare it to …, then …
When you subtract it from …, then …
When you write it as an equivalent, then …

Riddles for Grades 6–8

Similar styles of riddles might be proposed for students at these levels, except that the topics addressed would be more appropriate for their grade levels.

Some examples:

I’m thinking of a number more than one million but less than 10 million.
It has 4 zero digits.
It is not a multiple of 100.
It is a multiple of 6.
What might it be?

Students should realize that since the number is not a multiple of 100, the tens digit cannot be 0.
The only other digit that cannot be 0 is the millions digit. Therefore, we can have a 0 in the hundred thousands, ten thousands, thousands, hundreds, and ones digits.

The number could be 7 000 032 or 2 200 080 since these are multiples of 6.

I’m thinking of a two numbers.
They both round down to the nearest million.
They both round up to the nearest thousand.
They both round to the same value when rounding to the nearest ten thousand.
What might the two numbers be?

Students should realize that if they round down to the nearest million, the hundred thousands digit must be 4 or less.

If they round up to the nearest thousand, the hundreds digit should be 5 or greater.

If both numbers round to the same ten thousand value, the millions, hundred thousands, and ten thousands digit should be the same, and both thousands digits should be either 5 or more or both should be less than 5.

So, the numbers might be 1 432 517 and 1 431 815, or they might be 2 278 753 and 2 276 753.

I’m thinking of an integer.
When I subtract it from –10, I get an integer between 0 and –10.
When I add it to 30, I get an integer greater than 20.
What could my integer be?

Students should realize that the integer must be between –10 and 0 for the subtraction to work. They should realize that if they add an integer between –10 and 0 to 30, the value must be between 20 and 30, so as long the integer is 0, –1, –2, …, –9, it would work.

I’m thinking of a fraction.
When you multiply by \(\frac{5}{8}\), you get a result greater than \(\frac{1}{2}\).
When you divide \(\frac{12}{5}\) by that same fraction, the quotient is greater than 2\(\frac{1}{2}\).
What might the fraction be?

Students should realize that if you multiply \(\frac{5}{8}\) by a fraction and the result is most of the \(\frac{5}{8}\), you multiplied by a fraction less than 1 but pretty close to 1.

You might try \(\frac{3}{4}\).
\(\frac{5}{8}\) × \(\frac{3}{4}\) = 1532, but that’s not greater than \(\frac{1}{2}\).
So, you have to go higher.
You might try \(\frac{4}{5}\).
\(\frac{5}{8}\) × \(\frac{4}{5}\) is exactly one half, so you must go higher.
You might try \(\frac{9}{10}\).
\(\frac{5}{8}\) × \(\frac{9}{10}\) = \(\frac{45}{88}\), which is \(\frac{9}{16}\), and that is greater than \(\frac{1}{2}\).
And if you divide 125 by \(\frac{9}{10}\), you get \(\frac{24}{10}\) ÷ \(\frac{9}{10}\) = \(\frac{24}{9}\), and that is more than 2\(\frac{1}{2}\).
So, 910 is a possible fraction.

I’m thinking about the square root of a number.
The square root is between 2000 and 3000 less than the number that it is a square root of.
What could the number be?

Students should realize that the square root of a big number is a very small portion of it, so the number might be close to 2000.

You could try 2000. The square root is close to 45, so the difference is a bit under 2000. That is a bit too low.

You could try 2500. The square root is 50, and the difference is 2450, so that works.

There are lots of other possibilities; for example, 3000 – \sqrt{3000} is about 2945, so that works too.

I’m thinking of a repeating decimal.
When I write it as a fraction, the denominator is 90.
What might the decimal be?

Students should realize that, for example, 0.222 … is \(\frac{2}{9}\).
But you want 90ths. One way to do this is to find an equivalent fraction to \(\frac{2}{9}\) that has a denominator of 90; that would require multiplying the numerator and denominator of \(\frac{2}{9}\) by 10 to get \(\frac{20}{90}\). In this case, the decimal remains 0.222… .

You could also add a fraction with a denominator of 10 to a fraction with a denominator of 9, to end up with 90ths.

For example, add \(\frac{7}{10}\) to \(\frac{2}{9}\) to get \(\frac{83}{90}\).
The decimal format would then be 0.7 + 0.222 222… = 0.922 222 … .

There are other possibilities; for example, adding \(\frac{2}{10}\) to \(\frac{4}{9}\) to get \(\frac{58}{90}\).
The decimal would be 0.2 + 0.444 444… = 0.644 444 4 … .

Creating Their Own Riddles

You might encourage students to create their own riddles based on working with representations of fractions, decimals, integers, or large numbers, and using various operations.

Again, as with primary students, you might even suggest styles of clues, for example:
When you write the number …
When you add it to …, then …
When you show it as a decimal, then …
When you compare it to …, then …
When you subtract it from …, then …
When you write it as a square root and compare it to …, then …

In Summary

As you can see, it is relatively easy to come up with lots of riddles.
Not only are they attractive to students and exercise conceptual understanding, but they practise skills at the same time.