Project Euler/40/Solution - Revision history

Unknown user: /* Calculating the Product */

2025-10-04T16:42:55Z

Calculating the Product

← Older revision		Revision as of 16:42, 4 October 2025
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	<math>P = 1 \times 1 \times 5 \times 3 \times 7 \times 2 \times 1</math>		<math>P = 1 \times 1 \times 5 \times 3 \times 7 \times 2 \times 1</math>

	~~<math>P = 1 \times 5 \times 3 \times 7 \times 2</math>~~

	~~<math>P = 5 \times 3 \times 7 \times 2</math>~~

	~~<math>P = 15 \times 7 \times 2</math>~~

	<math>P = 105 \times 2</math>		<math>P = 105 \times 2</math>

Unknown user at 02:19, 20 April 2025

2025-04-20T02:19:22Z

← Older revision		Revision as of 02:19, 20 April 2025
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	* <math>C_6 = 5888889</math>		* <math>C_6 = 5888889</math>

	== 1. Finding the Digits ==		=== Finding the Digits ===

	* '''Finding <math>d_1</math>'''		* '''Finding <math>d_1</math>'''
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	:<math>d_{1000000} = 1</math>		:<math>d_{1000000} = 1</math>

	== 2. Calculating the Product ==		=== Calculating the Product ===

	We need to compute the product <math>P</math>:		We need to compute the product <math>P</math>:

Unknown user at 02:18, 20 April 2025

2025-04-20T02:18:55Z

← Older revision		Revision as of 02:18, 20 April 2025
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			{{Main\|Project Euler/40}}

			==Solution==

	Here is a calculation of the product <math>d_1 \times d_{10} \times d_{100} \times d_{1000} \times d_{10000} \times d_{100000} \times d_{1000000}</math>, where <math>d_n</math> is the <math>n</math>-th digit of the Champernowne constant <math>C_{10} = 0.123456789101112...</math>.		Here is a calculation of the product <math>d_1 \times d_{10} \times d_{100} \times d_{1000} \times d_{10000} \times d_{100000} \times d_{1000000}</math>, where <math>d_n</math> is the <math>n</math>-th digit of the Champernowne constant <math>C_{10} = 0.123456789101112...</math>.

Unknown user: /* 2. Calculating the Product */

2025-04-20T02:17:34Z

2. Calculating the Product

← Older revision		Revision as of 02:17, 20 April 2025
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	The product of the specified digits is 210.		The product of the specified digits is 210.

			==Flags==

			{{ProjectEulerFlag}}

Unknown user: /* 2. Calculating the Product */

2025-04-20T02:17:16Z

2. Calculating the Product

← Older revision		Revision as of 02:17, 20 April 2025
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	We need to compute the product <math>P</math>:		We need to compute the product <math>P</math>:

	<math>P = d_1 \times d_{10} \times d_{100} \times d_{1000} \times d_{10000} \times d_{100000} \times d_{1000000}</math>		<math>P = d_1 \times d_{10} \times d_{100} \times d_{1000} \times d_{10000} \times d_{100000} \times d_{1000000}</math>

	<math>P = 1 \times 1 \times 5 \times 3 \times 7 \times 2 \times 1</math>		<math>P = 1 \times 1 \times 5 \times 3 \times 7 \times 2 \times 1</math>

	<math>P = 1 \times 5 \times 3 \times 7 \times 2</math>		<math>P = 1 \times 5 \times 3 \times 7 \times 2</math>

	<math>P = 5 \times 3 \times 7 \times 2</math>		<math>P = 5 \times 3 \times 7 \times 2</math>

	<math>P = 15 \times 7 \times 2</math>		<math>P = 15 \times 7 \times 2</math>

	<math>P = 105 \times 2</math>		<math>P = 105 \times 2</math>

	<math>P = 210</math>		<math>P = 210</math>

	The product of the specified digits is 210.		The product of the specified digits is 210.

Unknown user: Created page with "Here is a calculation of the product $d_1 \times d_{10} \times d_{100} \times d_{1000} \times d_{10000} \times d_{100000} \times d_{1000000}$ , where $d_n$
2025-04-20T02:15:56Z

Created page with "Here is a calculation of the product <math>d_1 \times d_{10} \times d_{100} \times d_{1000} \times d_{10000} \times d_{100000} \times d_{1000000}</math>, where <math>d_n</math..."

New page
Here is a calculation of the product <math>d_1 \times d_{10} \times d_{100} \times d_{1000} \times d_{10000} \times d_{100000} \times d_{1000000}</math>, where <math>d_n</math> is the <math>n</math>-th digit of the Champernowne constant <math>C_{10} = 0.123456789101112...</math>.

We use the established method involving cumulative digit counts (<math>C_k</math>) for blocks of <math>k</math>-digit numbers.
* <math>C_1 = 9</math>
* <math>C_2 = 189</math>
* <math>C_3 = 2889</math>
* <math>C_4 = 38889</math>
* <math>C_5 = 488889</math>
* <math>C_6 = 5888889</math>

== 1. Finding the Digits ==

* '''Finding <math>d_1</math>'''
The first digit (<math>n=1</math>) is the first digit of '1'.
:<math>d_1 = 1</math>

* '''Finding <math>d_{10}</math>'''
:<math>n=10</math>. Falls into the 2-digit block (<math>k=2</math>), since <math>C_1 < 10 \le C_2</math>.
:Position within block: <math>m = n - C_1 = 10 - 9 = 1</math>.
:Integer index (<math>0</math>-based): <math>num\_index = \lfloor (m-1)/k \rfloor = \lfloor (1-1)/2 \rfloor = 0</math>.
:Integer <math>N = 10^{k-1} + num\_index = 10^{2-1} + 0 = 10</math>.
:Digit position within <math>N</math> (<math>1</math>-based): <math>digit\_pos = (m-1) \pmod{k} + 1 = (1-1) \pmod 2 + 1 = 1</math>. The 1st digit of 10.
:<math>d_{10} = 1</math>

* '''Finding <math>d_{100}</math>'''
:<math>n=100</math>. Falls into the 2-digit block (<math>k=2</math>), since <math>C_1 < 100 \le C_2</math>.
:Position within block: <math>m = 100 - C_1 = 100 - 9 = 91</math>.
:Integer index: <math>num\_index = \lfloor (91-1)/2 \rfloor = \lfloor 90/2 \rfloor = 45</math>.
:Integer <math>N = 10^{2-1} + 45 = 10 + 45 = 55</math>.
:Digit position: <math>digit\_pos = (91-1) \pmod 2 + 1 = 90 \pmod 2 + 1 = 1</math>. The 1st digit of 55.
:<math>d_{100} = 5</math>

* '''Finding <math>d_{1000}</math>'''
:<math>n=1000</math>. Falls into the 3-digit block (<math>k=3</math>), since <math>C_2 < 1000 \le C_3</math>.
:Position within block: <math>m = 1000 - C_2 = 1000 - 189 = 811</math>.
:Integer index: <math>num\_index = \lfloor (811-1)/3 \rfloor = \lfloor 810/3 \rfloor = 270</math>.
:Integer <math>N = 10^{3-1} + 270 = 100 + 270 = 370</math>.
:Digit position: <math>digit\_pos = (811-1) \pmod 3 + 1 = 810 \pmod 3 + 1 = 1</math>. The 1st digit of 370.
:<math>d_{1000} = 3</math>

* '''Finding <math>d_{10000}</math>'''
:<math>n=10000</math>. Falls into the 4-digit block (<math>k=4</math>), since <math>C_3 < 10000 \le C_4</math>.
:Position within block: <math>m = 10000 - C_3 = 10000 - 2889 = 7111</math>.
:Integer index: <math>num\_index = \lfloor (7111-1)/4 \rfloor = \lfloor 7110/4 \rfloor = 1777</math>.
:Integer <math>N = 10^{4-1} + 1777 = 1000 + 1777 = 2777</math>.
:Digit position: <math>digit\_pos = (7111-1) \pmod 4 + 1 = 7110 \pmod 4 + 1 = 2 + 1 = 3</math>. The 3rd digit of 2777.
:<math>d_{10000} = 7</math>

* '''Finding <math>d_{100000}</math>'''
:<math>n=100000</math>. Falls into the 5-digit block (<math>k=5</math>), since <math>C_4 < 100000 \le C_5</math>.
:Position within block: <math>m = 100000 - C_4 = 100000 - 38889 = 61111</math>.
:Integer index: <math>num\_index = \lfloor (61111-1)/5 \rfloor = \lfloor 61110/5 \rfloor = 12222</math>.
:Integer <math>N = 10^{5-1} + 12222 = 10000 + 12222 = 22222</math>.
:Digit position: <math>digit\_pos = (61111-1) \pmod 5 + 1 = 61110 \pmod 5 + 1 = 0 + 1 = 1</math>. The 1st digit of 22222.
:<math>d_{100000} = 2</math>

* '''Finding <math>d_{1000000}</math>'''
:<math>n=1000000</math>. Falls into the 6-digit block (<math>k=6</math>), since <math>C_5 < 1000000 \le C_6</math>.
:Position within block: <math>m = 1000000 - C_5 = 1000000 - 488889 = 511111</math>.
:Integer index: <math>num\_index = \lfloor (511111 - 1) / 6 \rfloor = \lfloor 511110 / 6 \rfloor = 85185</math>.
:Integer <math>N = 10^{6-1} + 85185 = 100000 + 85185 = 185185</math>.
:Digit position: <math>digit\_pos = (511111 - 1) \pmod 6 + 1 = 511110 \pmod 6 + 1 = 0 + 1 = 1</math>. The 1st digit of 185185.
:<math>d_{1000000} = 1</math>

== 2. Calculating the Product ==

We need to compute the product <math>P</math>:
<math>P = d_1 \times d_{10} \times d_{100} \times d_{1000} \times d_{10000} \times d_{100000} \times d_{1000000}</math>
<math>P = 1 \times 1 \times 5 \times 3 \times 7 \times 2 \times 1</math>
<math>P = 1 \times 5 \times 3 \times 7 \times 2</math>
<math>P = 5 \times 3 \times 7 \times 2</math>
<math>P = 15 \times 7 \times 2</math>
<math>P = 105 \times 2</math>
<math>P = 210</math>

The product of the specified digits is 210.