A number system discovered by archaeologists in a prehistoric cave indicates that the caveman used a number system that has 5 distinct shapes ∑, ∆, >, Ω and ↑. Furthermore it has been determined that the symbols ∑ to ↑ represents the decimal equivalents 0 to 5 respectively.

Centuries ago a caveman returning after a successful hunting expedition records his successful hunt on the cave wall by carving out the numbers ∆↑. What does the number ∆↑ represent? The below table indicates that the Caveman numbers ∆↑ represents decimal number 9.

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The Caveman is using a Base-5 number system. A Base-5 number system has five unique symbols representing numbers 0 to 4. To represent numbers larger than 4, a combination of 2, 3, 4 or more combinations of Caveman numbers are used. Therefore, to represent the decimal number 5, a two number combination of the Caveman number system is used. The most significant digit is ∆ which is equivalent to decimal 1. The least significant digit is ∑ which is equivalent to decimal 0. The five combinations of Caveman numbers having the most significant digit ∆, represent decimal values 5 to 9 respectively. This is similar to the Decimal Number system, where a 2-digit combination of numbers is used to represent values greater than 9. The most significant digit is set to 1 and the least significant digit varies from 0 to 9 to represent the next 10 values after the largest single decimal number digit 9.

The Caveman number ∆↑ can be written in expression form based on the Base value 5 and weights 5^0 , 5^1 , 5^2 etc.

= ∆ x 5^1 ↑ x 5^0

= ∆ x 5 + ↑ x 1

Replacing the Caveman numbers ∆ and ↑ with equivalent decimal values in the expression

yields.

= ∆ x 5^1 ↑ x 5^0

= 1 x 5 + 4 x 1 = 9

The number ∆Ω↑∑ in decimal is represented in expression form as:

= ∆ x 5^3 Ω x 5^2 ↑ x 5^1 ∑ x 5^0

= ∆ x 125 + Ω x 25 + ↑ x 5 + ∑ x 1

Replacing the Caveman numbers with equivalent decimal values in the expression yields.

= (1) x 125 + (3) x 25 + (4) x 5 + (0) x 1 = 125 + 75 + 20 + 0 = 220

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