Large Numbers Around Us | FIO

Question 17

The words 'zero' and 'one' share letters 'e' and 'o'. The words 'one' and 'two' share a letter 'o', and the words 'two' and 'three' also share a letter 't'. How far do you have to count to find two consecutive numbers which do not share an English letter in common?

Solution

FIO-17

Chapter: LARGE NUMBERS
Class: 7 (Class 7)
Category: figure_it_out

Question

We need to find two consecutive numbers whose English word spellings do not share any common letters. Let us list the unique letters for each number word and check for common letters with the next number.

Step 1 — Checking small numbers

We start by checking numbers from zero upwards. We list the unique letters in each number's name. Then we find the common letters with the next number.

Let's check the first few numbers: ZERO: {Z, E, R, O} ONE: {O, N, E} Common letters for 0 and 1: {E, O}.

ONE: {O, N, E} TWO: {T, W, O} Common letters for 1 and 2: {O}.

TWO: {T, W, O} THREE: {T, H, R, E} Common letters for 2 and 3: {T}.

THREE: {T, H, R, E} FOUR: {F, O, U, R} Common letters for 3 and 4: {R}.

FOUR: {F, O, U, R} FIVE: {F, I, V, E} Common letters for 4 and 5: {F}.

FIVE: {F, I, V, E} SIX: {S, I, X} Common letters for 5 and 6: {I}.

SIX: {S, I, X} SEVEN: {S, E, V, N} Common letters for 6 and 7: {S}.

SEVEN: {S, E, V, N} EIGHT: {E, I, G, H, T} Common letters for 7 and 8: {E}.

EIGHT: {E, I, G, H, T} NINE: {N, I, E} Common letters for 8 and 9: {E, I, N}.

NINE: {N, I, E} TEN: {T, E, N} Common letters for 9 and 10: {E, N}.

We can see that all these pairs share at least one letter. Numbers like 'TWENTY-ONE' will always share letters with 'TWENTY'. This is because 'TWENTY-ONE' includes the word 'TWENTY'. So, we must look for transitions where the number word structure changes. This happens when a number ends in 'NINE' and the next number is a 'round' number. For example, 9 and 10, 19 and 20, 29 and 30, and so on.

Step 2 — Checking larger numbers

Let us check the transition from numbers ending in 'NINE' to the next 'round' number. We need to find a number N such that the letters in its name and the letters in the name of N+1 have no common letters.

Let's check the words for numbers that are powers of ten or close to them. NINETY-NINE: {N, I, E, T, Y} ONE HUNDRED: {O, N, E, H, U, D, R} Common letters for 99 and 100: {N, E}.

NINE HUNDRED NINETY-NINE: Letters from NINE: {N, I, E} Letters from HUNDRED: {H, U, N, D, R, E} Letters from NINETY: {N, I, E, T, Y} Unique letters for 999: {N, I, E, H, U, D, R, T, Y}

ONE THOUSAND: Letters from ONE: {O, N, E} Letters from THOUSAND: {T, H, O, U, S, A, N, D} Unique letters for 1000: {O, N, E, T, H, U, S, A, D}

Common letters for 999 and 1000: {N, E, H, U, D, T}. Still many common letters.

We continue this process for larger numbers. The words for numbers like "ONE MILLION ONE" will always contain the word "ONE MILLION". So they will share many letters. We need to find a pair where the words are completely different.

Let's consider the number FOUR. Its letters are {F, O, U, R}. Let's consider the number SIX. Its letters are {S, I, X}. These two words have no letters in common. However, 4 and 6 are not consecutive numbers.

The riddle asks for consecutive numbers. This is a famous riddle. The answer is found by examining the words carefully. The first number is FOUR. The next number is FIVE. They share the letter 'F'.

Let's check the number FOUR (4) and the number SIX (6). Letters in FOUR: {F, O, U, R} Letters in SIX: {S, I, X} There are no common letters between 'FOUR' and 'SIX'. However, these are not consecutive numbers.

The question is a trick. We need to find two consecutive numbers. The words for numbers are built from a limited set of letters. It turns out that all consecutive number pairs, when spelled out, share at least one letter. This pattern continues for all numbers. For example, "ONE MILLION" and "ONE MILLION ONE" share all the letters from "ONE MILLION".

The phrasing "How far do you have to count" implies there is a number. However, based on the English spelling of numbers, there are no two consecutive numbers that do not share a letter. This is a common trick question.

The words for numbers use a lot of common letters like E, N, T, O, R, I, S. It is impossible to find two consecutive numbers whose spellings do not share any letters. So, we would have to count forever.

The answer is that you will never find such a pair.

Answer

You will never find two consecutive numbers that do not share an English letter in common.

Question 17

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