DuckieDai Pick's Theorem Python Program

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geometry · MIT License

Overview

Solve a lattice polygon's area, interior-point count, or boundary-point count with Pick's theorem.

Usage

Inputs

Requested unknown: area, interior points, or boundary points
Nonnegative interior count and positive boundary count where applicable

Outputs

Requested Pick's-theorem quantity or invalid-result warning

Units: Area is in square coordinate units; point counts are whole numbers.

Assumptions and limitations

Assumptions

The polygon is a lattice polygon and Pick's theorem applies.

Constraints

The TI program validates nonnegative or positive whole-number point counts.

Known failures

Non-lattice or inapplicable polygons produce an incorrect interpretation.

Compatibility and review

Review status: Tested on a TI calculator
Tested on: TI-84 Plus CE Python; OS 5.8.3; Python App 5.8.3.0048; exact source owner-tested through TI Connect CE
What happened: Owner confirmed the supplied Pick's theorem program runs on the physical calculator.
Dependencies: No imports
Suggested calculator name: PICKTHE — you can give it another valid, unique name when you transfer it
Desktop test cases: Not available for this interactive-only source

This exact source auto-launches an interactive calculator session, so compatibility evidence comes from the recorded physical-device review rather than an importable desktop fixture.

Browser preview

Try Pick's theorem

This browser preview calculates area from interior and boundary point counts. The TI program also solves the inverse counts.

Simulation boundary: This preview uses a reviewed browser adapter. It does not execute the downloaded Python and does not prove TI-Python or calculator compatibility.

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Exact reviewed bytes

Source code

Download .py

View source below, copy it when JavaScript is available, or download the exact .py bytes.

"""DuckieDai Pick's Theorem for TI-84 Plus CE Python."""


def duckiedai_intro(program_name, wait=True):
    """Show the reusable DuckieDai program opening."""
    title = program_name[:16]
    empty = 16 - len(title)
    left = empty // 2
    right = empty - left

    print("+----------------------+")
    print("|   " + " " * left + title + " " * right + "   |")
    print("|                      |")
    print("|       __             |")
    print("|   ___(o )>  quack!   |")
    print("|   \\ <_. )           |")
    print("|    `---'             |")
    print("|     by DuckieDai     |")
    print("+----------------------+")
    print("Loading...")

    if wait:
        input("Press enter to start ")
        print("\n" * 6)


def whole_number(prompt):
    """Require a nonnegative whole number."""
    while True:
        try:
            value = int(input(prompt))

            if value >= 0:
                return value

        except ValueError:
            pass

        print("Enter a whole number")
        print("0 or greater.")


def positive_whole(prompt):
    """Require a positive whole number."""
    while True:
        try:
            value = int(input(prompt))

            if value > 0:
                return value

        except ValueError:
            pass

        print("Enter a whole number")
        print("above 0.")


def positive_number(prompt):
    """Require a positive number."""
    while True:
        try:
            value = float(input(prompt))

            if value > 0 and value < 1e100:
                return value

        except ValueError:
            pass

        print("Enter a number above 0.")


def main():
    duckiedai_intro("PICKS THEOREM")

    print("PICK'S THEOREM")
    print()
    print("A = i + b/2 - 1")
    print()
    print("i = interior points")
    print("b = boundary points")
    print()
    print("What do you need?")
    print()
    print("1. Area")
    print("2. Interior points")
    print("3. Boundary points")
    print()

    choice = input("Choose 1-3: ")

    print()

    # ==========================================
    # 1. SOLVE AREA
    #
    # A = i + b/2 - 1
    # ==========================================

    if choice == "1":

        i = whole_number(
            "Interior points: "
        )

        b = positive_whole(
            "Boundary points: "
        )

        area = (
            i +
            b / 2 -
            1
        )

        print()
        print("Using:")
        print("A = i + b/2 - 1")
        print()
        print("A = " + str(i))
        print("+ " + str(b) + "/2")
        print("- 1")
        print()

        print("ANSWER")
        print("Area = " + str(area))
        print("square units")


    # ==========================================
    # 2. SOLVE INTERIOR POINTS
    #
    # i = A - b/2 + 1
    # ==========================================

    elif choice == "2":

        area = positive_number(
            "Area: "
        )

        b = positive_whole(
            "Boundary points: "
        )

        i = (
            area -
            b / 2 +
            1
        )

        print()
        print("Using:")
        print("A = i + b/2 - 1")
        print()
        print("Rearranged:")
        print("i = A - b/2 + 1")
        print()

        if i < 0 or i != int(i):

            print("INVALID RESULT")
            print()
            print("Interior points")
            print("must be a whole")
            print("number.")
            print()
            print("Check your values.")

        else:

            i = int(i)

            print("ANSWER")
            print("Interior points =")
            print(str(i))


    # ==========================================
    # 3. SOLVE BOUNDARY POINTS
    #
    # b = 2(A - i + 1)
    # ==========================================

    elif choice == "3":

        area = positive_number(
            "Area: "
        )

        i = whole_number(
            "Interior points: "
        )

        b = (
            2 *
            (area - i + 1)
        )

        print()
        print("Using:")
        print("A = i + b/2 - 1")
        print()
        print("Rearranged:")
        print("b = 2(A-i+1)")
        print()

        if b <= 0 or b != int(b):

            print("INVALID RESULT")
            print()
            print("Boundary points")
            print("must be a positive")
            print("whole number.")
            print()
            print("Check your values.")

        else:

            b = int(b)

            print("ANSWER")
            print("Boundary points =")
            print(str(b))

    else:
        print("Invalid choice.")

    print()
    print("DuckieDai says: done!")


# TI-Python imports the selected AppVar rather than
# setting __name__ to main.
main()

"""DuckieDai Pick's Theorem for TI-84 Plus CE Python."""
def duckiedai_intro(program_name, wait=True):
"""Show the reusable DuckieDai program opening."""
title = program_name[:16]
empty = 16 - len(title)
left = empty // 2
right = empty - left
print("+----------------------+")
print("| " + " " * left + title + " " * right + " |")
print("| |")
print("| __ |")
print("| ___(o )> quack! |")
print("| \\ <_. ) |")
print("| `---' |")
print("| by DuckieDai |")
print("+----------------------+")
print("Loading...")
if wait:
input("Press enter to start ")
print("\n" * 6)
def whole_number(prompt):
"""Require a nonnegative whole number."""
while True:
try:
value = int(input(prompt))
if value >= 0:
return value
except ValueError:
pass
print("Enter a whole number")
print("0 or greater.")
def positive_whole(prompt):
"""Require a positive whole number."""
while True:
try:
value = int(input(prompt))
if value > 0:
return value
except ValueError:
pass
print("Enter a whole number")
print("above 0.")
def positive_number(prompt):
"""Require a positive number."""
while True:
try:
value = float(input(prompt))
if value > 0 and value < 1e100:
return value
except ValueError:
pass
print("Enter a number above 0.")
def main():
duckiedai_intro("PICKS THEOREM")
print("PICK'S THEOREM")
print()
print("A = i + b/2 - 1")
print()
print("i = interior points")
print("b = boundary points")
print()
print("What do you need?")
print()
print("1. Area")
print("2. Interior points")
print("3. Boundary points")
print()
choice = input("Choose 1-3: ")
print()
# ==========================================
# 1. SOLVE AREA
#
# A = i + b/2 - 1
# ==========================================
if choice == "1":
i = whole_number(
"Interior points: "
)
b = positive_whole(
"Boundary points: "
)
area = (
i +
b / 2 -
1
)
print()
print("Using:")
print("A = i + b/2 - 1")
print()
print("A = " + str(i))
print("+ " + str(b) + "/2")
print("- 1")
print()
print("ANSWER")
print("Area = " + str(area))
print("square units")
# ==========================================
# 2. SOLVE INTERIOR POINTS
#
# i = A - b/2 + 1
# ==========================================
elif choice == "2":
area = positive_number(
"Area: "
)
b = positive_whole(
"Boundary points: "
)
i = (
area -
b / 2 +
1
)
print()
print("Using:")
print("A = i + b/2 - 1")
print()
print("Rearranged:")
print("i = A - b/2 + 1")
print()
if i < 0 or i != int(i):
print("INVALID RESULT")
print()
print("Interior points")
print("must be a whole")
print("number.")
print()
print("Check your values.")
else:
i = int(i)
print("ANSWER")
print("Interior points =")
print(str(i))
# ==========================================
# 3. SOLVE BOUNDARY POINTS
#
# b = 2(A - i + 1)
# ==========================================
elif choice == "3":
area = positive_number(
"Area: "
)
i = whole_number(
"Interior points: "
)
b = (
2 *
(area - i + 1)
)
print()
print("Using:")
print("A = i + b/2 - 1")
print()
print("Rearranged:")
print("b = 2(A-i+1)")
print()
if b <= 0 or b != int(b):
print("INVALID RESULT")
print()
print("Boundary points")
print("must be a positive")
print("whole number.")
print()
print("Check your values.")
else:
b = int(b)
print("ANSWER")
print("Boundary points =")
print(str(b))
else:
print("Invalid choice.")
print()
print("DuckieDai says: done!")
# TI-Python imports the selected AppVar rather than
# setting __name__ to main.
main()

Filename: PICKTHE.py
Size: 5044 bytes
SHA-256: 1714c126a432524b92b65e5458d8a81699a3015d89d725f9ebc2295a23eca18a

Transfer and launch

Confirm that your calculator is the Python-capable model named above. Check that its OS and Python App versions match the reviewed record.
Download the .py source above and verify its SHA-256 digest if your computer provides that option.
Use TI Connect CE to send the Python file to a compatible calculator. PICKTHE is a suggested name; you may choose another valid, unique calculator name.
Open the Python App, select the program, and check sample inputs before relying on other results.

Read the complete installation, launch, troubleshooting, and removal guide.

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Version history

1.0.0 — current version, published 2026-09-24. Download this reviewed version

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