What number base is this?

Logic Level 3

228 , 72 , 53 , 34 , ? \LARGE \color{#D61F06} {228},\color{#69047E}{72},\color{#20A900} {53},\color{magenta} {34},?


The answer is 25.

Parth Lohomi by Parth Lohomi , 20, India

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2 solutions

Shubhendra Singh
May 6, 2015

Here's the logic

  • 2 2 + 2 2 + 8 2 = 72 2^{2}+2^{2}+8^{2}=72

  • 7 2 + 2 2 = 53 7^{2}+2^{2}=53

  • 5 2 + 3 2 = 34 5^{2}+3^{2}=34

  • ? = 3 2 + 4 2 = 25 ?=3^{2}+4^{2}=\large{25}

8 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

Good. Can the same rules applies to the preceding terms? That is, does there exist an integer before the number 228 228 ? If yes, can you find the smallest possible number before 228 228 ? And the smallest number before it?

Since there is the four square theorem, there will always be a preceding term (a 4-digit one at least) for any number in the sequence.

Karan Jain - 6 years, 1 month ago

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@ C h a l l e n g e m a s t e r : @Challenge master: The previous number before 228 228 can be 6888 , 8688 , 8868 , 8886 , 7779 , 7797 , 7977 , 9777 6888,8688,8868,8886,7779,7797,7977,9777 smallest of which is 6888 6888

Vighnesh Raut - 6 years, 1 month ago
Arpit MIshra
May 6, 2015

Each succeeding number is equal to the sum of squares of the digits of the previous number:

If a number is = 29 \huge\color{#333333}{29} the next number will be equal to ( 2 2 + 9 2 ) = 85 \huge\color{#333333}(2^{2}+9^{2}) = 85

  • 2 2 + 2 2 + 8 2 = 4 + 4 + 64 = 72 \huge\color{#D61F06}{2^{2}+2^{2}+8^{2} = 4 + 4 + 64 }= \color{#69047E}{72}

  • 7 2 + 2 2 = 49 + 4 = 53 \huge\color{#69047E}{7^{2}+2^{2} = 49 + 4 }= \color{#20A900}{53}

  • 5 2 + 3 2 = 25 + 9 = 34 \huge\color{#20A900}{5^{2}+3^{2} =25 + 9}= \color{magenta}{34}

  • 3 2 + 4 2 = 9 + 16 = 25 \huge\color{magenta}{3^{2}+4^{2} = 9 + 16 }= \color{turquoise}{25}

The next numbers in this series are, 228 , 72 , 53 , 34 , 25 , 29 , 85 , 89 , 145 , 42 , 20 , 4 , 16 , 37 , 58 , 89 , 145 , 42 , 20 , 4 , 16 , 37 , 58 , . . . . \huge \color{#D61F06} {228},\color{#69047E}{72},\color{#20A900} {53},\color{magenta} {34}, \color{turquoise}{25}, \color{#333333}{29},\color{maroon}{85},\color{#302B94}{89},\color{lime}{145},\color{rosybrown}{42},\color{cyan}{20},\color{khaki}{4},\color{teal}{16},\color{olive}{37}, \color{plum}{58}, \color{#302B94}{89}, \color{lime}{145},\color{rosybrown}{42},\color{cyan}{20},\color{khaki}{4},\color{teal}{16},\color{olive}{37}, \color{plum}{58}, \color{#333333}{. . . .}

This is an i n f i n i t e \huge infinite series since there are infinite preceding terms and the series goes in a loop after the number 58 \huge \color{plum}{58} .

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

Nice. Are there infinite number of distinct integers in this sequence? If no, can you find the sum of all distinct numbers in this sequence? If yes, why?

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