Why Is 29 Special?

 This little puzzle appeared in Puzzle A Day.

What is the only number that, when written in typed letters as all capitals, is made up of exactly the same number of straight-line segments as the value of the number itself? For example, FIVE uses 10 straight lines, so it cannot be the answer. (Ignore any hyphens in the spelled-out number.)

I asked Gemini to create a table for the number of straight line segments in each letter of the English alphabet and then solve the puzzle.

Python

def number_to_words(n):
    # Basic implementation for 0-99
    # Not handling large numbers as we expect small solutions
    ones = ["ZERO", "ONE", "TWO", "THREE", "FOUR", "FIVE", "SIX", "SEVEN", "EIGHT", "NINE"]
    teens = ["TEN", "ELEVEN", "TWELVE", "THIRTEEN", "FOURTEEN", "FIFTEEN", "SIXTEEN", "SEVENTEEN", "EIGHTEEN", "NINETEEN"]
    tens = ["", "", "TWENTY", "THIRTY", "FORTY", "FIFTY", "SIXTY", "SEVENTY", "EIGHTY", "NINETY"]
    
    if 0 <= n < 10:
        return ones[n]
    elif 10 <= n < 20:
        return teens[n-10]
    elif 20 <= n < 100:
        if n % 10 == 0:
            return tens[n // 10]
        else:
            return tens[n // 10] + " " + ones[n % 10]
    return "NUMBER_TOO_LARGE"

# Letter segment counts
# Assumption: Sans-serif, standard straight lines.
# B: 1 vertical
# D: 1 vertical
# G: 1 horizontal
# J: 0 (hook) or 1 (vertical)? Let's assume 0 first as usually curve.
# P: 1 vertical
# Q: 1 tail
# R: 2 (vertical + diagonal)
# U: 0 (curve)
# Y: 3
segments = {
    'A': 3, 'B': 1, 'C': 0, 'D': 1, 'E': 4, 'F': 3, 'G': 1, 'H': 3, 'I': 1, 
    'J': 0, 'K': 3, 'L': 2, 'M': 4, 'N': 3, 'O': 0, 'P': 1, 'Q': 1, 'R': 2, 
    'S': 0, 'T': 2, 'U': 0, 'V': 2, 'W': 4, 'X': 2, 'Y': 3, 'Z': 3
}

# Check matches
matches = []
for i in range(100):
    word = number_to_words(i)
    # Remove spaces and hyphens if any (my function adds space)
    clean_word = word.replace(" ", "").replace("-", "")
    count = 0
    for char in clean_word:
        if char in segments:
            count += segments[char]
    
    if count == i:
        matches.append((i, word, count))

print("Matches:", matches)

Code output

Matches: [(16, 'SIXTEEN', 16), (29, 'TWENTY NINE', 29)]

Based on a standard analysis of the straight line segments in the uppercase English alphabet (using a typical sans-serif block font), here is the breakdown.

Table of Straight Line Segments per Letter

Letter	Segments	Shape Breakdown
A	3	Two diagonal, one horizontal
B	1	One vertical (curves do not count)
C	0	All curved
D	1	One vertical
E	4	One vertical, three horizontal
F	3	One vertical, two horizontal
G	1	One short horizontal shelf
H	3	Two vertical, one horizontal
I	1	One vertical
J	0	Typically a hook (though some fonts use 1 vertical)
K	3	One vertical, two diagonal
L	2	One vertical, one horizontal
M	4	Two vertical, two diagonal
N	3	Two vertical, one diagonal
O	0	All curved
P	1	One vertical
Q	1	One short diagonal tail
R	2	One vertical, one diagonal leg
S	0	All curved
T	2	One vertical, one horizontal
U	0	Typically a curve (block versions may have 2 vertical)
V	2	Two diagonal
W	4	Four diagonal
X	2	Two diagonal
Y	3	Two diagonal, one vertical (or three meeting at center)
Z	3	Two horizontal, one diagonal

Nano Bananas attempt to encapsulate the information

The Numbers That Match

Using the values from the table above, we can deduce which integers (when written as English words) have a value equal to their total number of straight line segments.

There are exactly two integers that satisfy this condition:

1. SIXTEEN

• S (0) + I (1) + X (2) + T (2) + E (4) + E (4) + N (3)

• Calculation: $0 + 1 + 2 + 2 + 4 + 4 + 3 = \mathbf{16}$

• Result: The word "SIXTEEN" is made of 16 straight lines, which equals its value.

2. TWENTY NINE

• TWENTY: T (2) + W (4) + E (4) + N (3) + T (2) + Y (3) = 18

• NINE: N (3) + I (1) + N (3) + E (4) = 11

• Calculation: $18 + 11 = \mathbf{29}$

• Result: The words "TWENTY NINE" are made of 29 straight lines, which equals its value.
  
  

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In the answer supplied to the puzzle, only 29 is listed and not 16 as well. Why I'm not sure but the letters of 29 are made up ONLY of straight line segments whereas 16 contains the curved letter S.