The paradox—

My mate and I from home did start,
Some little space we were apart.
When we had run a mile or more
We kept our distance, as before;
Shade of Colenso! could this be,
When twice as fast as I ran he?

is solved by the fact that the lines apply to the large and small wheels of a bicycle.

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The enigma from Lewis Carroll’s Papers—

A monument all men agree
Am I in all sincerity,
Half cat, half hindrance made.
If head and tail removed should be,
Then most of all you strengthen me.
Replace my head, the stand you see
On which my tail is laid.

is solved by Tablet.

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