[LS036] How Many Ones?

036 - How Many Ones?

How Many Ones? is a puzzle in Professor Layton and the Last Specter. The puzzle must be solved in order to progress the story.

Puzzle

US Version

One day, a little boy who was very curious about numbers thought to himself: "If I were to write out all the numbers from 1 to 120, how many times would I write the numeral 1?"

Can you work out the answer?

UK Version

One day, a little boy who was very curious about numbers thought to himself: "If I were to write out all the numbers from 1 to 120, how many times would the number 1 appear?"

Can you work out the answer?

Hints

Click a Tab to reveal the Hint.

Try writing the numbers down and counting the 1s.

US Version

Be careful with numbers with more than one 1, such as 11.

UK Version

Watch out for numbers with more than one 1, such as 11.

The single digits are easy.
There's just one 1.

Think about the double digits.

US Version

There's one 1 in 10 and two 1s in 11. There's also one 1 in every number from 12 to 19, for a total of eight. There are eight more in 21, 31, 41, 51, 61, 71, 81, and 91.

UK Version

There's one 1 in 10 and two 1s in 11. There's also one 1 in every number from 12 to 19, giving a total of eight 1s. There are eight more in 21, 31, 41, 51, 61, 71, 81 and 91.

What about the triple digits?

There's a single 1 in 100, and two 1s in both 101 and 110.

US Version

111 has three 1s, and there are two 1s in every number from 112 to 119. Are there any numbers you haven't considered yet? Don't forget about the 1 in 120, either.

UK Version

111 has three 1s, and there are two 1s in every number from 112 to 119. There's also a single 1 in 120.

Now, are there any numbers you haven't considered yet?

Solution

Incorrect

US Version

Too bad!

Are there any 1s that you missed?

UK Version

Too bad.

Did you maybe forget to count some 1s?

Correct

US Version

Precisely!

The numeral 1 comes up 53 times.

You have to remember to count it twice in 11 and three times in 111. If you keep this in mind, the problem is simple.

UK Version

One-derful!

The number 1 comes up 53 times.

You have to remember to count it twice in 11 and three times in 111. Keep this in mind and the problem is simple.

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