diff --git a/ch02/05_bpe-from-scratch/bpe-from-scratch.ipynb b/ch02/05_bpe-from-scratch/bpe-from-scratch.ipynb
index 18d3d91..9853ac4 100644
--- a/ch02/05_bpe-from-scratch/bpe-from-scratch.ipynb
+++ b/ch02/05_bpe-from-scratch/bpe-from-scratch.ipynb
@@ -861,7 +861,7 @@
    "metadata": {},
    "source": [
     "- Next, let's initialize and train the BPE tokenizer with a vocabulary size of 1,000\n",
-    "- Note that the vocabulary size is already 255 by default due to the byte values discussed earlier, so we are only \"learning\" 745 vocabulary entries \n",
+    "- Note that the vocabulary size is already 256 by default due to the byte values discussed earlier, so we are only \"learning\" 744 vocabulary entries (if we consider the `<|endoftext|>` special token and the `Ġ` whitespace token; so, that's 742 to be precise)\n",
     "- For comparison, the GPT-2 vocabulary is 50,257 tokens, the GPT-4 vocabulary is 100,256 tokens (`cl100k_base` in tiktoken), and GPT-4o uses 199,997 tokens (`o200k_base` in tiktoken); they have all much bigger training sets compared to our simple example text above"
    ]
   },
@@ -908,7 +908,7 @@
    "id": "36c9da0f-8a18-41cd-91ea-9ccc2bb5febb",
    "metadata": {},
    "source": [
-    "- This vocabulary is created by merging 742 times (~ `1000 - len(range(0, 256))`)"
+    "- This vocabulary is created by merging 742 times (`= 1000 - len(range(0, 256)) - len(special_tokens) - \"Ġ\" = 1000 - 256 - 1 - 1 = 742`)"
    ]
   },
   {
@@ -975,12 +975,12 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[424, 256, 654, 531, 302, 311, 256, 296, 97, 465, 121, 595, 841, 116, 287, 466, 256, 326, 972, 46, 256, 60, 124, 271, 683, 102, 116, 461, 116, 124, 62]\n"
+      "[424, 256, 654, 531, 302, 311, 256, 296, 97, 465, 121, 595, 841, 116, 287, 466, 256, 326, 972, 46, 60, 124, 271, 683, 102, 116, 461, 116, 124, 62]\n"
      ]
     }
    ],
    "source": [
-    "input_text = \"Jack embraced beauty through art and life. <|endoftext|> \"\n",
+    "input_text = \"Jack embraced beauty through art and life.<|endoftext|> \"\n",
     "token_ids = tokenizer.encode(input_text)\n",
     "print(token_ids)"
    ]
@@ -1000,7 +1000,7 @@
     }
    ],
    "source": [
-    "input_text = \"Jack embraced beauty through art and life. <|endoftext|> \"\n",
+    "input_text = \"Jack embraced beauty through art and life.<|endoftext|> \"\n",
     "token_ids = tokenizer.encode(input_text, allowed_special={\"<|endoftext|>\"})\n",
     "print(token_ids)"
    ]
@@ -1015,7 +1015,7 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Number of characters: 57\n",
+      "Number of characters: 56\n",
       "Number of token IDs: 21\n"
      ]
     }