I got only two correct answers for this puzzle – from Suman Saraf and Pratik Poddar – thank you both and well done!
I am taking the liberty of copying the answer from Pratik – I find it to be very succinctly articulated.
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From square marked 1 to square marked 64, the maximum manhattan distance possible between the two squares is 14 and the maximum difference between any two numbers is 63. Since 63/14>4, there has to be at least one step where the jump is greater than 4. Hence proved.
Hope you all enjoyed the puzzle!