There are several reports of Instructables.com users encountering reCAPTCHA errors while trying to post new instructables, or even a bug report of the problem on the forums and sometimes even comments.

I got an "Unmatching Captcha Values" error while I tried to post a new instructable.
Determined to find the cause of the problem, I did some digging using the Inspect Element tool on Google Chrome.
It revealed that the problem of Captcha not loading is due to an outdated URL which causes a 404 error.

After a few experiments, I've found a quick fix solution for this problem until the root cause (captcha script URL 404 error) is fixed by Instructables. These steps apply to Google Chrome users, although it should be possible using similar tools available in other browsers such as Firefox.

Step 1: Fix ReCAPTCHA - Use Inspect Element Tool

Get to the publish page, or whichever page where the captcha error occurs.

Locate the text that says "This helps us reduce spam". The captcha is supposed to appear above this text.
Right click on the text and select the "Inspect element" option.

Step 2: Fix ReCAPTCHA - Inspect Element Console

In the Inspect Element window, goto the Console tab.

You will see a "Uncaught ReferenceError: Recaptcha is not defined" error message, which indicates the problem with the reCAPTCHA script. This is caused due to a bad url in the page - http://api.recaptcha.net/js/recaptcha_ajax.js . Google has updated the reCAPTCHA and moved the location of the script to http://www.google.com/recaptcha/api/js/recaptcha_ajax.js .

Open a new tab on Google Chrome and goto the URL: http://www.google.com/recaptcha/api/js/recaptcha_ajax.js

Select the entire content of the page (use Ctrl+A) and copy it.

Come back to the Inspect Element console and paste the contents in the console and press Enter.

Step 3: Fix ReCAPTCHA - Locate and Copy Script

Goto the Elements tab in the Inspect Element console.
Locate the </form> tag in the HTML code shown.
Just below the </form> tag you will see <script type="text/javascript"> ... </script>.
Expand the script tag, by clicking the arrow next to it.
You will see a Recaptcha.create function with a text like this:

Recaptcha.create(
          "6LcUzgIAAAAAALMl6Ab36B3L29jzk-FPYAUJDz49",
          "recaptchaContainer",
          {
            theme: "red",
            callback: function() {
              var recaptchaInput = $("recaptcha_response_field");
              recaptchaInput.addClassName("required");
              recaptchaInput.setStyle({"width":"310px"});
              recaptchaInput.tabIndex = 8;
            }
          }
        );

Copy that text.

Step 4: Fix ReCAPTCHA - Done !

Go back to the Console tab in Inspect Element.
Paste the script you copied earlier.
Press Enter.

The reCAPTCHA should now appear on the page.
Close the Inspect Element window, enter the captcha as shown.

Thats it !